6
6
Probability
PROBABILITY
The Lotto game in NSW has been going strong since 1979 and is now the most popular form of lottery. It it a game of chance in which 6 balls are selected from a barrel of coloured balls numbered 1 to 44. Players who predict the correct 6 numbers share a first prize pool of at least 1 million dollars. Every Australian state has its own version of Lotto, and there are also national versions such as Powerball, OzLotto and the Pools. The original Lotto game used only 40 balls, but by adding 4 new balls in 1989 the chance of winning first prize dropped from 1 in 3.8 million to 1 in 7 million. How did the addition of 4 extra balls make this popular game almost twice as difficult to win?
This chapter is about the mathematics of chance and counting. As well as revising probability theory from the Preliminary Course, you will learn about the principles of counting and listing all possible outcomes of multistage events, especially ones involving arrangements and selections. You will use probability concepts and techniques to evaluate the fairness of games of chance, overcome common gambling misconceptions, and make informed decisions involving probability.
In this chapter you will learn how to:
n construct and use tree diagrams to list the sample space for multistage events
n count the number of outcomes for a multistage event by multiplying the number of choices at each stage
n count the number of ways in which different items can be arranged
n count the number of ordered and unordered selections of a given size that can be made from a group of different items
n calculate probabilities involving ordered and unordered selections
n construct and use probability tree diagrams to solve multistage probability problems
n calculate the expected number of times a particular outcome should occur over a number of trials, and the financial expectation of a game of chance
n carry out probability simulations to model events and analyse their results
THE MEANING OF PROBABILITY
This section revises probability theory and ideas introduced in the Preliminary Course. n An outcome is a result of an experiment or game. For example, when two coins are tossed
together, one possible outcome is HT (head–tail).
n The sample space is the set of all possible outcomes. For example, the sample space when two coins are tossed together has four outcomes: {HH, HT, TH, TT}.
n An event is a group of one or more outcomes. For example, the event of both coins being the same consists of the two outcomes {HH, TT}.
n The theoretical probability of an event, E, occurring is calculated by the following formula:
Example 1
One card is selected at random from a normal deck of playing cards. What is the probability that the selected card is:
(a) an ace? (b) a red ace?
(c) a picture card? (d) a club or a red ace?
Solution
A normal deck contains 52 cards (4 suits of 13 cards each). The sample space has 52 outcomes: n(S) = 52.
(a) P(ace) = = 4 aces in a deck
(b) P(red ace) = = 2 red aces (heart and diamond)
(c) P(picture card) = = 4 jacks, 4 queens and 4 kings = 12
(d) P(club or red ace) = = 13 clubs + 2 red aces
P(E) = =
where n(E) is the number of favourable outcomes in the event
n(S) is the total number of outcomes in the sample space.
number of favourable outcomes total number of outcomes
--- n E( )
n S( ) ---2 2 A A 3 3 6 6 5 5 4 4 J J Q Q K K 9 9 10 1 0 8 8 7 7 2 2 A A 4 4 3 3 J J Q Q K K 9 9 10 1 0 8 8 7 7 6 6 5 5 J J Q Q K K 8 8 6 6 7 7 4 4 5 5 3 3 2 2 A A 10 1 0 9 9 A A 2 2 3 3 J J K K Q Q 7 7 6 6 5 5 4 4 9 9 10 1 0 8 8 4 52 --- 1 13 ---2 52 --- 1 26 ---12 52 --- 3 13
---13+2 52 --- 15
The range of probabilities
A probability value can be expressed as a fraction or decimal, ranging from 0 to 1, or as a percentage from 0 to 100%.
Complementary events
If P(E) is the probability of E occurring, then P( ) is the probability of Enot occurring. is called the complementary event, and P(E) +P( ) = 1.
Example 2
A jar contains 12 red, 7 yellow, 8 white and 13 black jellybeans. Express as a decimal the probability that a jellybean randomly selected from this jar is:
(a) green (b) black or white (c) not red
(d) black, yellow or red (e) not blue
Solution
Total number of jellybeans = 12 + 7 + 8 + 13 = 40
(a) P(green) = = 0 No green jellybeans, impossible event
(b) P(black or white)= = = 0.525 (c) P(not red)= 1 −P(red)
= 1 − = = 0.7
(d) P(black, yellow or red)=
= = 0.8 or P(black, yellow or red)= 1 − P(white)
= 1 − = = 0.8
(e) P(not blue) = = 1 All not blue, certain event 0 < P(E) < 1
If P(E) = 0, then event E is impossible (‘will never happen’). If P(E) = 1, then event E is certain (‘must happen’).
E˜
E˜ E˜
P( )= 1 − P(E)
P(event does not occur)= 1 − P(event does occur)
E˜
0 40
---13+8 40
---21 40
---12 40
---28 40
---13+7+12 40
---32 40
---8 40
---32 40
---Experimental probability
The experimental probability of an event occurring is its relative frequency.
Example 3
A Roulette wheel at a casino has 37 numbers: 0 to 36. The results of 250 spins of the wheel are shown in the table.
Express your answers to the following questions as percentages (correct to 1 decimal place where appropriate).
(a) What is the experimental probability (relative frequency) of spinning a number from 19 to 27?
(b) What is the calculated probability (theoretical probability) of spinning a number from 19 to 27?
(c) What is the calculated probability of spinning 0? (d) What is the experimental probability of spinning 0?
(e) What is the experimental probability of spinning a number less than 10?
Solution
Total frequency = 5 + 60 + 62 + 64 + 59 = 250
(a) Experimental probability (19–27)= × 100% From table
≈ 25.6%
(b) P(19–27)= × 100% 9 numbers out of 37
≈ 24.3%
(c) P(0)= × 100% 1 number out of 37
≈ 2.7%
(d) Experimental probability (0)= × 100% From table
= 2%
(e) Experimental probability (,10)= × 100% From table
= 26%
Outcome Frequency
0 5
1–9 60
10–18 62
19–27 64
28–36 59
Relative frequency of an event = frequency of the event total frequency
---64 250
---9 37
---1 37
---5 250
---1. From the letters of the name NEW CENTURY, one is selected at random. Find the probability that it is:
(a) N (b) Y (c) a consonant
(d) E or U (e) not T (f) A
2. Kevin buys 8 tickets in a raffle that sells a total of 1250 tickets. Calculate the probability that he wins first prize, expressing the answer as:
(a) a decimal (b) a percentage
3. A survey was taken on the types of vehicles passing an intersection over a 1-hour period. Calculate correct to 3 decimal places the probability that the next vehicle to pass the intersection is:
(a) a bus (b) a truck (c) not a car
(d) neither a truck nor a semi-trailer
4. Lisa spins the wheel at right to determine her prize. What is the probability that she wins:
(a) the CDs?
(b) the cash or the dinner? (c) the computer?
(d) neither the holiday nor the computer?
5. A traffic light is green for 60 seconds, amber for 4 seconds and red for 80 seconds. Calculate the percentage probability (correct to 1 decimal place) that a traffic light will show red.
6. A piggy bank has 4 $2 coins, 8 $1 coins, 3 50-cent coins and 6 20-cent coins. One coin is taken out at random. What is the probability that it is:
(a) a 20-cent coin? (b) a gold coin?
(c) not a $2 coin? (d) not a 20-cent coin?
7. In a batch of 160 instant (scratch) lottery tickets, 35 of them contain a cash prize. What is the probability of winning a cash prize on an instant lottery ticket from this batch?
8. (a) A card is selected at random from a normal deck of 52 cards. What is the probability that the card selected is:
(i) a diamond? (ii) a red 10? (iii) black?
(b) Suppose the card selected was black. Now a second card is selected from the remaining cards. What is the probability that this card also is black?
9. What is the probability of not having been born on a Tuesday?
10. There are 7 people on the school council: Kate, Marco, Amy, Naomi, Megan, Ahmed and Ben. One of them is randomly selected to represent the school at the mayor’s lunch. What is the probability that the person selected is:
(a) female? (b) not Naomi or Megan?
(c) a person whose name begins with A? (d) a person with an A in his or her name?
Exercise 6-01:
The meaning of probability
Vehicle Frequency
Car 722
Truck 132
Semi-trailer 58
Motor bike 34
Bus 19
CDs
Holiday
Dinner
Computer
11. A road survey shows the number of car accidents occurring on each day of the week.
Calculate as percentages (correct to 3 significant figures) the probability that a car accident occurs on:
(a) a Monday (b) a Friday
(c) a day other than Thursday (d) the weekend
12. Jodie’s English class has 22 students. If 4 students are chosen at random from this class to make a speech, what is the probability that Jodie will be chosen?
13. Four cards marked with the numbers 1, 2, 3 and 4 are placed face down on a table. One card has been turned over as shown.
What is the probability that the next card turned over has an even number on it?
14. In a football match, the Knights have a 42% chance of beating the Dragons while the Dragons have a 51% chance of beating the Knights.
(a) What other outcome is possible? (b) What is the probability of this outcome?
15. A bag contains 16 red, 19 blue and 15 yellow balls. If 1 ball is drawn out at random, what is the probability that it is:
(a) blue? (b) not yellow? (c) not green?
(d) neither red nor blue? (e) green? (f) white, red or yellow?
16. Give an example of an event that has a probability of:
(a) 1 (b) 0
17. The stand-by times of a batch of mobile phone batteries were tested and recorded. Calculate the percentage probability that a battery selected at random will have a standby time of:
(a) 70–74 hours (b) 60–69 hours (c) 75 or more hours (d) 79 or fewer hours
18. If this spinner is spun, what is the probability that the arrow will land on:
(a) an even number? (b) a number greater than 5? (c) a 5?
(d) a 6?
19. There are 5 people in a group. What is the probability that one (or more) of them has a birthday on 15 April?
Day Mon Tue Wed Thu Fri Sat Sun
No. of accidents 6 4 8 14 28 18 18
3
Stand-by time
(hours) Frequency
60–64 4
65–69 8
70–74 134
75–79 11
80–84 3
7
4
2 4 6
10
6
20. (a) What is the probability that a telephone number selected at random from the phone book ends in 9?
(b) Why isn’t this the same as the probability that the number selected begins with 9?
21. A die is loaded so that the chance of 6 appearing is double the chance of any one of the 5 other numbers. Find the probability of rolling:
(a) a 6 (b) a 5 or 6
(c) an odd number (d) a number less than 6
22. A box contains r red, w white and b brown socks. Write an expression for the probability of randomly selecting a sock that is:
(a) white (b) not red
TREE DIAGRAMS AND TABLES
Using a tree diagram
A tree diagram is a convenient way of listing all of the possible outcomes of a multistage event. A multistage event consist of two or more events occurring together, such as rolling two 6s on a pair of dice or getting rain 3 days in a row.
Example 4
A coin is tossed 3 times. Find all of the possible outcomes (of the sample space): (a) by listing them (b) by using a tree diagram
Solution
Let H = head, T = tail.
(a) {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} 8 possible outcomes
(b)
A tree diagram is a systematic way of listing all the outcomes of a sample space because it ensures that all possible arrangements have been covered. It is useful for counting multistage events because it uses branches to illustrate the possibilities at every stage or level. The coin tosses above are an example of a 3-stage experiment, with each toss being a different stage.
Example 5
The digits 7, 2, 3 and 6 are written on separate cards and two of them are selected at random to form a 2-digit number.
(a) Use a tree diagram to list all possible outcomes.
(b) What is the probability that the number formed is divisible by 3? H
T
H
T
H
T
H
T
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT H
T
H
T H
T
Solution
(a)
(b) There are 12 possible outcomes, and 4 numbers that are divisible by 3 (ticked above).
P(number divisible by 3) = =
Using a table
The outcomes of a sample space can also be found using a table.
Example 6
A pair of dice are rolled and the sum of the numbers is calculated. Use a table to list all possible outcomes and hence find the probability of rolling a sum of 10.
Solution
There are 36 possible outcomes.
P(sum of 10) = = 3 ways of getting a sum of 10
Note: The sample space of this 2-stage event could also have been found using a tree
diagram, but the table is more organised and concise.
1. (a) A family has 4 children. List the possible arrangements of boys and girls. (b) What is the probability that the family consists of:
(i) 3 girls and 1 boy? (ii) 2 girls and 2 boys?
2. In the dice game Craps, a pair of dice are rolled and the total is calculated. A player wins if he rolls a 7 or 11, and loses if he rolls a 2 or 12. Calculate the probability of:
(a) a win (b) a loss (c) rolling a total greater than 9
2nd die
1st die
+ 1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
2 3 6
7 3 6
7 2 6
7 2 3
72 ✔
73 76
27 ✔
23 26
37 32 36 ✔
67 62 63 ✔
7
2
3
6
1st digit 2nd digit Outcomes
Note that if a number is used as the first digit, it cannot be used as the second digit as well.
4 12 --- 1
3
---3 36 --- 1
12
3. The numbers 2, 5, 6 and 9 are written on separate cards and placed in a box. Two cards are selected at random, one after the other, to form a 2-digit number. Draw a tree diagram to list all possible outcomes, then calculate the probability that the number formed is:
(a) 65 (b) odd
(c) less than 59 (d) divisible by 5
4. Three coins are tossed together. Find the probability of getting:
(a) exactly 1 head (b) 2 heads and 1 tail (in that order) (c) 2 heads and 1 tail (in any order) (d) no heads
5. Erin, Kristen, Shelley, Adelle and Bianca are in a basketball team. They need to select a captain, then a vice-captain.
(a) Draw a tree diagram to determine how many pairings are possible. (b) If every player is equally likely to be picked, what is the probability that:
(i) Kristen and Adelle fill the positions (in any order)? (ii) Kristen is captain and Adelle is vice-captain? (iii) Erin is captain?
6. Over the 4 days of the Easter long weekend (Friday to Monday), on any day the probability that it will rain is 50%.
(a) Use a tree diagram to list and count all possibilities of rain (R) and no rain ( ) over the 4 days.
(b) What is the probability that it will rain on exactly 2 of the days? (c) What is the probability that it will rain on at least 1 of the days?
7. A coin and a die are thrown together. Find the probability of throwing: (a) a head and an odd number
(b) a tail and a number greater than 2
8. A, C and T are written on separate cards and drawn out of a box in random order. (a) Use a tree diagram to list and count all the possible ways the cards can be drawn out. (b) Hence calculate the probability that:
(i) the letters are drawn out in the order CAT (ii) A is the last letter drawn out
(iii) T is not the last letter drawn out
9. When Mr Faulds drives to work he needs to pass through 4 sets of traffic lights. Each light has an equal chance of showing red or green (count amber as red).
(a) Use a tree diagram to determine how many different arrangements of red and green lights are possible for the 4 sets.
(b) Calculate the probability that in his journey to work Mr Faulds faces: (i) all green lights (ii) 2 red lights (in any order) (iii) at least 1 red light (iv) 3 green lights
10. A pair of dice are rolled and the difference between the higher and lower numbers is calculated. Use a table to find all 36 possible outcomes.
(a) What is the probability of rolling a difference of 2?
(b) What is the probability of rolling a difference that is greater than 2? (c) Which difference has the highest probability?
11. (a) How many different 3-digit numbers can be formed from the digits 5, 9, 8 and 1 if each digit can be used only once?
(b) What is the probability that one of these 3-digit numbers selected at random is: (i) greater than 500? (ii) even? (iii) divisible by 3?
Adam and Eve play a game in which they each hold out one hand and show some number of fingers (1 to 5) to the other person at the same time. If the total number of fingers shown on both hands is even, Adam wins. If the total is odd, Eve wins.
1. Is this a fair game?
2. If you were Adam, how could you improve your chances of winning?
3. If you were Eve, how could you improve your chances of winning?
THE MULTIPLICATION PRINCIPLE FOR COUNTING
For multistage probability problems, it is often impractical to list all possible outcomes using a tree diagram because the total number of possibilities may be too large. In these cases, we use more advanced counting techniques.
Example 7
From these lists of given names and surnames, determine how many different arrangements of given name–surname pairs are possible.
Solution
Starting a tree diagram but not finishing it, we get:
For each given name, there are 5 possible surnames. Number of possible arrangements = 6 × 5 = 30
Example 8
From this menu, calculate how many different 3-course dinners are possible.
Entrée Pumpkin soup Calamari rings Potato wedges
Main course Cajun prawns Steak diane Roast lamb Chicken dijon
Grilled perch
Dessert Pavlova Black forest cake Chocolate mousse Mangoes and ice cream
Group modelling activity:
A handy odds and evens game
Given names Alex Brionne
Cate Daniel
Erin Fiona
Surnames Garrett
Hijazi Iacano Johnson
Kee
Garrett Hijazi Iacano Johnson Kee Garrett Hijazi Iacano . . . etc. Alex
Brionne
Cate
Daniel
Erin
Fiona
Solution
There are 3 possible entrées, 5 possible main courses, 4 possible desserts. Number of possible 3-course dinners = 3 × 5 × 4 = 60
Example 9
An Internet user password is made up of 6 characters, alphabetic or numeric (e.g. D5RE21). How many different passwords are possible?
Solution
Possible characters at each stage = 26 letters + 10 digits = 36
Number of possible passwords = 36 × 36 × 36 × 36 × 36 × 36 = 366 = 2 176 782 336
6 characters means 6 stages.
For the above example, if lower-case letters (a, b, c) are counted as being different from upper-case letters (A, B, C), how many more passwords are possible?
1. Anthony has 10 shirts, 6 pairs of trousers and 4 pairs of shoes. In how many different ways can he dress himself?
2. A poker machine has 5 wheels. Each wheel can show 12 different pictures. How many arrangements of 5 pictures are possible on one line?
3. A school needs to elect a boy captain and a girl captain. The candidates are Carlos, David, John, Lydia, Vicki, Martin, Peter and Rosa.
(a) How many pairings of boy and girl captains are possible? (b) List them.
4. In a football-tipping competition, a player must pick the winners of 7 football matches each week. In each match, either of the 2 teams can win (assume no draws). How many possible outcomes are there for the 7 football matches?
The multiplication principle for counting arrangements is:
1. If A can be arranged in m ways and B in n ways, then A and B together can be arranged in m × n ways.
2. More generally, if A can be arranged in a ways, B can be arranged in b ways,
C can be arranged in c ways, etc., then A, B, C, … together can be arranged in a × b × c × … ways
Think:
Even more passwords
5. Telephone numbers in North Coast NSW have 8 digits, but the first 2 digits must be 66. How many possible telephone numbers are there for this region?
6. At a school camp, students are allowed to choose one activity for each of the three sessions during the day.
Over one day, how many different schedules of activities are possible?
7. In the game of Yahtzee, 5 dice are rolled together. How many different outcomes are possible?
8. FM radio stations in NSW have a call sign of 2 followed by 3 letters of the alphabet (e.g. 2DAY). How many different call signs are possible using this system?
9. A new make of car has the following options:
Colour: white, red, blue, green or silver Transmission: manual or automatic
Air conditioning: yes or no Airbags: yes or no
Model: standard, deluxe or luxury How many different arrangements of options are available?
10. A bank account PIN (personal identification number) is made up of 4 digits. (a) How many possible PINs are there?
(b) If letters of the alphabet are used instead of numbers, how many possible 4-letter PINs are there?
11. A coin is tossed 10 times. How many arrangements of heads and tails are possible?
12. An electronic security gate is opened by a 6-character code where the first 2 characters are letters (from A to F) and the last 4 characters are digits (from 0 to 9) (e.g. DF1269). How many different codes are possible?
13. In Year 12, students at Westvale High School must study 6 subjects, one from each line. Line 1 contains only English, line 2 contains only Mathematics, and the electives for lines 3 to 6 are listed below.
How many subject selections are possible at Westvale High?
14. Ten friends—Raymond, Grace, Jacqueline, Nicole, Cindy, Andrew, Lisa, Harry, William and Ivy—go to a dance.
(a) How many boy–girl dance pairs are possible? (b) List them.
Morning Archery
Karate Rock climbing Bird watching Bushwalking
Gymnastics
Midday Swimming
Canoeing Ropes course
Cooking
Afternoon Orienteering
Flying fox Caving Horseriding
Fishing Obstacle course
Line 3 Computing Studies
Physics Ancient History
Line 4 Biology
Art Industrial Technology
Japanese Music
Line 5 PD Health PE
Chemistry Drama Business Studies
Ceramics
Line 6 Modern History
15. A family has 6 children. How many arrangements of boys and girls are possible?
16. In the Braille alphabet (for the blind), characters are represented by a cell of 1 to 6 raised dots printed in two columns. The cells for the letters Y, E and N are shown below.
How many different cells are possible using the Braille system? (Hint: A completely blank cell is not an option.)
17. In NSW, a black-on-white car registration plate is made up of 3 letters, 2 numbers and 1 letter, in that order (e.g. BQZ–35N). How many arrangements of this type of number plate are possible?
18. In the late 1990s, telephone numbers in Australia changed from 7 digits to 8 digits. (a) Why do you think this happened?
(b) How many extra phone numbers were possible by the addition of the extra digit?
19. In Morse code, letters are represented by a series of dashes (—) and dots (•). The number of dashes and dots in a code ranges from 1 to 4. For example, • is the code for E while — • — — is the code for Y. How many codes are possible using this system?
(Hint: Count the number of codes available using 1, 2, 3 and 4 characters.)
20. Computers use a code of 7 binary digits (0s and 1s) to represent characters. For example, 1010000 means P while 0110100 means 4. How many different characters can be represented by a code of 7 digits?
21. A coin is tossed n times. Write an expression for the number of possible arrangements of heads and tails.
COUNTING ARRANGEMENTS
Example 10
Six friends—Rachel, Ross, Chandler, Monica, Phoebe and Joey—stand in line for a group photo. How many possible arrangements of positions are there?
Solution
Draw boxes for the six positions.
There are 6 ways of arranging someone in the 1st position; but once that person is placed, there are only 5 ways of arranging someone else in the 2nd position, 4 ways of arranging someone else in the 3rd position, and so on, up to only 1 way of arranging the last person in the last position.
Number of possible arrangements = 6 × 5 × 4 × 3 × 2 × 1 = 720
Example 11
Harry’s office has 10 car spaces outside for his employees’ cars. In how many different ways can 10 cars be parked in 10 spaces?
Y E N
Solution
There are 10 ways of arranging a car in the 1st space, 9 ways of arranging another car in the 2nd space,
8 ways of arranging another car in the 3rd space, and so on. Number of possible ways = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3 628 800
Factorial notation (x!)
Did you notice the pattern in the answers to the two examples above?
In mathematics, the ! symbol (read factorial) is used to describe a special product. For example:
4!= 4 × 3 × 2 × 1 ‘4 factorial’
6!= 6 × 5 × 4 × 3 × 2 × 1 ‘6 factorial’ 10!= 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 ‘10 factorial’
x! means the product of all whole numbers from x down to 1. Your calculator even has an
key. Enter 10 to show that
10! = 3 628 800
Example 12
Suppose the 6 friends from Example 10 wish to have smaller group photos on a couch that seats only 4 people. Each photo will show only 4 people. How many different photos are possible? That is, in how many ways can you arrange 6 people into 4 positions?
Solution
Draw boxes for the 4 positions on the couch.
Number of possible arrangements = 6 × 5 × 4 × 3 = 360
Note that because there are only 4 positions, the multiplication does not go all the way back to 1, so the key on the calculator cannot be used.
Example 13
A girls’ school is electing a captain and a vice-captain. There are 5 candidates: Ang, Beth, Cassie, Dasha and Elena.
(a) How many possible pairings of captain/vice-captain are there? (b) List them.
Solution
(a)
Number of possible pairings = 5 × 4 = 20
10 9 8 7 6 5 4 3 2 1
x! x!
The number of ways in which n different items can be arranged is
n! = n × (n − 1) × (n − 2) × … × 2 × 1
6 5 4 3
x!
5 4
(b) The 20 possible pairings (captain, vice-captain) are:
(A, B) (A, C) (A, D) (A, E) A = Ang
(B, A) (B, C) (B, D) (B, E) B = Beth
(C, A) (C, B) (C, D) (C, E) C = Cassie
(D, A) (D, B) (D, C) (D, E) D = Dasha
(E, A) (E, B) (E, C) (E, D) E = Elena
1. Five boys—Arden, Dominic, Mark, Ivan and Brad—run a 100 m race. (a) How many 1st–2nd–3rd–4th–5th placings are possible?
(b) How many 1st–2nd–3rd placings are possible?
2. How many different 2-digit numbers can be made from the digits 3, 8 and 1 if: (a) the digits can be repeated? (b) the digits cannot be repeated? List the possibilities in both cases.
3. How many different ways are there of placing 4 letters into 4 envelopes (1 letter per envelope)?
4. In horse racing, an exacta bet is choosing the first 2 horses in a race in the correct order (1st–2nd).
(a) How many exacta bets are possible for a 19-horse race?
(b) Benny has 3 favourite horses in the race: Jasmine’s Joy, Mr Lucky and Big Shot. How many exacta bets can he make using his 3 favourites? List them.
5. Each ticket to a concert has a 3-letter code where the letters cannot be repeated. How many different tickets can be printed?
6. An exam consists of 12 multiple choice questions, where the correct answer could be A, B, C or D. How many different ways are there of answering all 12 questions?
7. In how many ways can 12 rowers sit in the 12 seats on a dragonboat?
8. In how many ways can 4 books (an atlas, French text, English novel and Maths dictionary) be arranged in a row on a shelf? List them.
9. In how many different ways can the letters of the word CENTURY be arranged?
10. From a netball team of 7, a captain and vice-captain need to be selected. How many different pairings are possible?
11. There are 24 horses in a race.
(a) A trifecta is a bet on the first 3 horses in a race, in the correct order (1st–2nd–3rd). How many trifectas are possible for this race?
(b) A superfecta is a bet on the first 6 horses in a race, in the correct order (1st–2nd– 3rd–4th–5th–6th). How many superfectas are possible for this race?
The number of ways in which n different items can be arranged in r positions is
n × (n − 1) × (n − 2) ×… [r terms]
For example, the number of ways in which 6 items can be arranged in 4 positions is 6 × 5 × 4 × 3 [4 terms]
12. How many different 3-digit numbers can be made from the digits 3, 8, 1 and 5 if: (a) the digits can be repeated?
(b) the digits cannot be repeated?
13. In how many different ways can Barry, Maurice and Robin sit in a row? List the ways.
14. A school council needs to elect a president, secretary and treasurer from its 13 members. In how many ways can the election turn out if each member has an equal chance?
15. Outside the town hall there are 3 flagpoles. The mayor’s office owns 8 different flags. In how many ways can the flags be arranged on the 3 flagpoles (1 flag per flagpole)?
16. Three girls—Marcia, Jan and Cindy—have a swimming race. (a) How many possible 1st–2nd–3rd placings are there? List them. (b) How many possible 1st–2nd placings are there? List them. (c) How many possible 1st placings are there? List them.
17. There are 4 seats in Kramer’s car: 2 at the front, 2 at the back.
(a) How many different ways are there of seating 4 people—Kramer, Jerry, George and Elaine—in the car? List them.
(b) List the different ways of filling the 2 back seats if Kramer drives?
18. How many different house numbers can be made from the digits 1, 4 and 6 if digits cannot be repeated? List them all.
19. (a) In how many different ways can 3 girls and 2 boys line up at the canteen? (b) In how many ways can they line up if the 3 girls must be ahead of the 2 boys?
20. Kate organises a household roster for her family so that every night one person washes up while another wipes up. There are 6 people in her family: Mum, Dad and 4 children. (a) How many pairings are possible?
(b) How many pairings are possible if Dad won’t wipe up?
(c) How many pairings are possible if Dad won’t wipe up and Mum won’t wash up?
Just for the record
O
NE-
ARMED BANDITSBefore poker machines became electronic and computerised, they were mechanical and each wheel was a spinning drum of different symbols. The wheels were all activated by pulling a large lever on the side of the machine with a knob on the end.
These days, poker machines are very popular in clubs and hotels, but they are there to earn money for the clubs and the state government. Although small wins occasionally occur, for every $100 a punter puts into a machine, he or she can expect to win between −$8 and −$15 (i.e. lose between $8 and $15) in the long run. In 1998–99, $5.4 billion were lost to gambling in NSW, more than half to poker machines. This represents an average of $1067 for every adult in NSW!
1. Why do you think Australians call poker machines ‘one-armed bandits’?
What is the largest number x! that can be calculated by your calculator? Why?
COUNTING UNORDERED SELECTIONS
Example 14
Six friends visit a tennis court. How many different doubles teams (2 players) are possible?
Solution
There are 6 × 5 = 30 possible arrangements, but this isn’t the answer to the problem. Let the 6 players be denoted by A, B, C, D, E, F. Then the 30 possibilities are:
AB AC AD AE AF
BA BC BD BE BF
CA CB CD CE CF
DA DB DC DE DF
EA EB EC ED EF
FA FB FC FD FE
But with a doubles team, AB and BA are the same team. So are AC and CA, etc. That means we have ‘double-counted’ the arrangements. This situation is an example of an unordered selection—the order is not important. The two players in a team can ‘swap places’ and it will still be the same team. Because of this, we must divide our answer by 2! or 2.
Number of possible doubles teams = = = 15
Example 15
Three students are to be selected from a group of 8 students to represent the school at a talented writers’ day. How many combinations of 3 students are possible?
Solution
This is another example of an unordered selection. Once the 3 students are selected, it doesn’t matter what order they are within the group. For example, for 3 students A, B, C,
ABC = ACB = BAC = BCA = … etc.
Number of possible combinations=
= =
= 56
Technology:
The largest factorial on your calculator
6 5
6×5 2×1 --- 30
2
---no. of arrangements of 3 students from 8 students no. of ways 3 students can rearrange themselves
---8×7×6 3×2×1 --- 336
6
---The number of unordered selections that can be made from n different items, when there are r positions, is
[r terms]
For example, the number of unordered selections that can be made from 6 items when there are 4 positions is
n×(n–1)×(n–2)×…
r!
---Example 16
In Lotto, 6 balls are randomly selected from a barrel of 44 numbered balls. How many different selections of 6 balls are possible?
Solution
There are 44 balls and 6 positions.
Number of possible selections = 6 terms in numerator and denominator
= The denominator is 6!
= 7 059 052
So there is about 1 chance in 7 million of winning Lotto if you enter one set of 6 numbers.
Example 17
A student council must elect 8 junior members and 4 senior members. There are 25 junior candidates and 11 senior candidates.
(a) How many different ways are there of electing the 8 junior members? (b) How many different ways are there of electing the 4 senior members? (c) Hence, how many different student councils are possible?
Solution
A council is an unordered selection.
(a) Junior members: 25 candidates, 8 positions
Number of possible ways=
=
= 1 081 575 (b) Senior members: 11 candidates, 4 positions
Number of possible ways=
= = 330
(c) Number of possible student councils= 1 081 575 × 330 = 356 919 750
1. Phillipa wants to study 2 languages at university. She can choose from Chinese, French, Spanish, Vietnamese and Polish.
(a) How many selections are possible? (b) List the selections.
2. After winning a radio competition, Matthew can select 10 CDs from a collection of 24. How many possible selections of CDs can he make?
3. Mrs Kapp asked her class of 24 students to work in groups of 3. How many different groups are possible?
44×43×42×41×40×39 6×5×4×3×2×1
---5 082 ---517 440 720
---25×24×23×22×21×20×19×18 8×7×6×5×4×3×2×1
---4.36×1010
40 320
---11×10×9×8 4×3×2×1
---7920 24
4. In horse racing, a quinella is betting on the first 2 horses in a race, in any order (1st–2nd or 2nd–1st). How many different quinella bets are possible for a 22-horse race?
5. In court, a jury panel of 12 members is to be made up from a group of 40 candidates. How many different panels are possible?
6. Before 1989, there were only 40 numbered balls in the NSW Lotto game. How many combinations of 6 balls were possible back then?
7. Bruno selects 10 cards from a normal deck of 52 cards. How many different selections are possible? Answer in scientific notation correct to 3 significant figures.
8. Kim has 3 free concert tickets which she wants to give to 3 of her 5 friends. (a) In how many ways can she give away these tickets?
(b) List the ways.
9. Four students need to be selected from the student council of 8 to represent the school at an ANZAC Day ceremony. How many selections of 4 representatives are possible?
10. There are 4 members in a debating team but only 3 of them speak. How many speaking teams of 3 are possible? List them.
11. A hospital’s casualty section is supervised by a team of 5 nurses at a time. If there are 15 nurses available, how many different teams are possible?
12. Svetlana has 8 coins in her purse and draws 3 of them out at random. How many different combinations are possible?
13. Six boys are playing handball but only 4 can play on the court at a time. How many groupings of 4 are possible:
(a) if the order on the court is not important?
(b) if the order on the court (who is in which square) is important?
14. In an English exam, Walid needs to answer 3 questions from section A and 2 questions from section B.
(a) How many possible selections of 3 questions can Walid make in section A if it contains 10 questions?
(b) How many possible selections of 2 questions can he make in section B if it contains 8 questions?
(c) Hence, in how many different ways can Walid answer this English exam?
15. In the Pools, 6 numbers are chosen from 1 to 38. How many possible selections are there?
16. A pet shop has 5 kittens for sale. In how many different ways can Patty select kittens if she is allowed to purchase:
(a) 2 kittens? (b) 1 kitten?
(c) 3 kittens? (d) 5 kittens?
17. In a lucky dip at the fair, box A contains 20 prizes while box B contains 14 prizes. (a) Lyle is allowed to choose 2 prizes from box A and 1 prize from box B. How many
different selections are possible?
18. A group of 20 students visit the ten-pin bowling centre and must organise themselves in groups of 6 per lane. How many groupings are possible?
19. How many different hands of 5 cards can be selected from all the hearts in a normal deck of cards?
20. A family has 6 children but only 2 are allowed to sit in the front of the family van. (a) How many ways are there of choosing 2 children from a group of 6?
(b) How many ways are there of choosing 4 children from a group of 6?
(c) What do you notice about your answers to parts (a) and (b)? Can you work out why?
Just for the record
L
OTTO-
MANIAEvery state of Australia has its own lotto game, and there are also national lotto games such as OzLotto, Powerball and the Pools. The NSW game is called Lotto and involves the selection of 6 numbers from 1 to 44. The Victorian game is called Tattslotto and involves the selection of 6 numbers from 1 to 45. Other states and OzLotto also use 45 numbers. Powerball uses two barrels of 45 numbers and chooses 5 numbers from one barrel and 1 number (‘the powerball’) from the other. This means the powerball could be the same as one of the numbers already chosen.
1. Is it harder to win Lotto or Powerball from a single entry?
2. Robert plays Lotto every week and uses the same six numbers: 1, 2, 3, 4, 5, 6. His mother tells him he’s crazy because ‘These numbers will never come up!’ What does she mean? Is she right? What is Robert’s strategy?
3. Investigate the different types of lotto games by visiting your local newsagent or the NSW Lotteries’ website: www.nswlotteries.com.au.
Study tips
G
ETTING HELPn Use your Maths teachers as resources. They should be your first port of call. They know your HSC Course best as well as your strengths and weaknesses.
n Use the library, Maths department or bookstore to find study guides, summaries and past exam papers (including HSC exams).
n Use your friends and classmates. Study in groups if you find it easier. Ask an older student for help.
n Use your family; it doesn’t matter if they don’t know the mathematics you’re studying. Give them your revision notes and ask them to quiz you. Sometimes, even explaining the work to someone else makes things clearer to yourself.
n Use the Internet. HSC resources can be found on websites run by the Board of Studies (www.boardofstudies.nsw.edu.au), the Department of Education (www.hsc.csu.edu.au) and the Mathematical Association of NSW (www.hsc.csu.edu.au/pta/mansw). n Use the HSC Advice Line. Before and during the HSC exam period, the Board of
ORDERED AND UNORDERED SELECTIONS
It is important to distinguish between the two types of selections met in this chapter.
Ordered selections (also called permutations) are arrangements in which the order within the group is important. There are more of them (compared to unordered selections) because ABC, ACB, BAC, etc. are considered different. Examples are:
n the first 3 placings in a horse race
n electing a president, secretary and treasurer of a committee n arranging photos on a page in a photo album.
Unordered selections (also called combinations) are arrangements in which the order within the group is not important. There are fewer of them (compared to ordered selections) because ABC, ACB, BAC, etc. are considered the same. Examples are:
n choosing 5 players for a basketball team n selecting 6 numbers for a Lotto game n selecting a sample of 20 items to be tested.
Example 18
Poker is a card game in which each player is dealt a ‘hand’ of 5 cards from a normal deck of 52 cards.
(a) How many different hands of 5 cards are possible?
(b) In how many ways can 3 aces be selected from the 4 aces in a deck? (c) In how many ways can 2 queens be selected from the 4 queens in a deck? (d) Hence, what is the probability of being dealt a hand of 3 aces and 2 queens?
Solution
With a Poker hand, the order of the cards dealt is not important. For example, K 2 7 A 5 = A 7 K 5 2 = ... etc. This is an example of an unordered selection (combination).
(a) Number of possible hands=
=
= 2 598 960
(b) Number of possible ways of selecting 3 aces from 4 aces = = 4
(c) Number of possible ways of selecting 2 queens from 4 queens = = 6 (d) Number of possible ways of selecting 3 aces and 2 queens = 4 × 6 = 24
Hence, P(3 aces, 2 queens) = = 1 chance in 108 290
Groupings
Arrangements
Arranging in order
Permutations
Ordered selections
Selections Combinations
Unordered selections
52×51×50×49×48 5×4×3×2×1
---311 875 200 120
---4×3×2 3×2×1
---4×3 2×1
---24 2 598 960 --- 1
---Example 19
Dad believes that in a 14-horse race, 3 of his 5 ‘favourites’ must come 1st, 2nd and 3rd (in any order). A trifecta is a bet on the first 3 places of a horse race, in the correct order. (a) How many trifectas are possible from 14 horses?
(b) If Dad wants to cover all trifectas of his 5 favourites, how many trifecta bets is this? (c) If all 14 horses are equally likely to win the race, what is the probability that Dad will
win from one of his bets?
Solution
A trifecta is an ordered selection (permutation) because the 1st–2nd–3rd order is important. (a)
Number of possible trifectas = 14 × 13 × 12 = 2184 (b)
Number of trifectas from 5 favourites = 5 × 4 × 3 = 60
(c) P(Dad wins) = =
Your calculator may have special keys for counting ordered and unordered selections. n For ordered selections, you can use the n_{P}
r (permutations) key.
Example: How many trifectas are possible from 14 horses?
14 horses, 3 positions, order important.
Enter 14 3 to get the answer 2184. n For unordered selections, you can use the n_{C}
r (combinations) key.
Example: How many Poker hands are possible from a deck of 52 cards?
52 cards, 5 positions, order not important.
Enter 52 5 to get the answer 2 598 960.
1. State whether each situation describes an ordered or unordered selection. (a) betting on the Melbourne Cup trifecta
(b) a car registration number plate
(c) selecting a volleyball team out of 14 candidates (d) using a 6-digit code to open a suitcase
(e) selecting a hand of 7 cards from a deck of cards (f) electing a student council of 8 students
(g) creating a sample of 20 light globes to test
2. In an election, the 5 candidates are listed at random on the ballot (voting) paper. (a) How many ways are there of listing the 5 candidates?
(b) What is the probability that the candidates are listed in alphabetical order?
(c) If the first candidate listed is Ms Ward, how many different ways are there of listing the other candidates?
14 13 12
1st 2nd 3rd
5 4 3
1st 2nd 3rd
60 2184 --- 5
182
---Technology:
Permutations and combinations on the calculator
n_{P} r =
n_{C} r =
3. Arthur has 4 Maths books and one English book. He wishes to arrange these books on a shelf.
(a) In how many possible ways can he do this?
(b) In how many of these ways do the 4 Maths books stay together?
4. A bag contains 6 red marbles and 4 blue marbles. Four marbles are drawn from the bag. (a) How many different selections are possible?
(b) How many different ways are there of selecting 3 red marbles (from 6) and 1 blue marble (from 4)?
(c) Hence, what is the probability of selecting 3 red and 1 blue marble from the bag?
5. From a group of 7 authors, 3 are selected to write a book. What is the probability that Klaas, David and Colin are selected?
6. (a) How many 4-letter arrangements can be made from the word COMPUTER? (b) What is the probability that one of the arrangements selected at random begins with
the letter P?
7. Jesse needs to visit 4 places today: the dentist, video store, solicitor and bank. (a) In how many different ways can he visit them today?
(b) If every selection is equally likely, what is the probability that he visits the dentist first?
8. Two students are selected at random from a group of 5 girls and 4 boys. (a) How many different selections are possible?
(b) How many of these selections have 2 girls?
(c) Hence, what is the probability that both of the students selected are girls?
9. Dana has 4 ice cream flavours in her fridge: chocolate, vanilla, mango and spearmint. (a) She wants to make a triple-decker ice cream cone of 3 flavours. How many possible
triple-decker cones are there?
(b) What is the probability that the chocolate scoop will be on top?
(c) Narelle, who is visiting Dana, prefers to eat her 3 flavours in a bowl, so the order within the bowl does not matter. Show that there are 4 possible combinations of flavours and list them.
(d) What is the probability that the chocolate scoop is not included in Narelle’s combination?
10. One boy and 2 girls sit in a row. What is the probability that the 2 girls sit together?
11. From a normal deck of cards, 4 cards are randomly selected. Calculate the probability that the hand contains:
(a) 4 queens (b) an ace, king, queen and jack
(c) the ace, king, queen and jack of clubs (d) one 5
(e) three 10s and an ace (f) 2 red cards and 2 black cards
12. (a) In Lotto, a player needs to select the 6 winning numbers from 1 to 44. How many possible outcomes are there?
(b) In Lotto Strike, a player needs to select the first 4 numbers of a Lotto draw in the correct order. How many possible outcomes are there?
(c) Which game is harder to win: Lotto or Lotto Strike?
14. A bag contains 10 $2 coins, 5 $1 coins and 15 10-cent coins. Melanie randomly selects 3 coins from the bag. What is the probability that she brings out:
(a) 3 $2 coins? (b) 1 of each coin? (c) 2 10-cent coins?
15. In Poker, a ‘royal flush’ is getting 10, J, K, Q and A of the same suit. Show that there is 1 chance in 649 740 of achieving this.
16. Brad and Janet book into separate rooms on the top floor of a small hotel. There are 6 vacant rooms and they are randomly assigned a room each.
(a) In how many different ways can Brad and Janet be assigned their rooms? (b) What is the probability that they are assigned rooms that are next to each other?
17. Five boys and 4 girls are at camp. Six of them are selected to ride in a rowboat. (a) In how many different ways can they be selected for the rowboat?
(b) What is the probability that the selected group contains 3 boys and 3 girls? (c) In how many different ways can they be selected if their order of seating in the
rowboat is important?
18. The letters of the word MATHS are randomly arranged. Calculate the probability that: (a) the letters are in alphabetical order
(b) the arrangement begins with M and ends with S (c) the arrangement begins with M or S
19. A poker machine has 4 wheels and each wheel can show 10, J, Q, K or A. (a) How many outcomes are possible?
(b) What is the probability of getting 4 aces in a row? (c) What is the probability of getting any 3 aces?
20. Five friends—Jack, Kevin, Lex, Marcel and Nick—play squash together. (a) List all of the possible ways 2 of them can be chosen to play each other first. (b) What is the probability that Nick plays in the first game?
(c) What is the probability that neither Kevin nor Lex plays in the first game?
(d) The remaining 3 players sit on a bench. What is the probability that they sit in their order of height?
PROBABILITY TREE DIAGRAMS
A probability tree diagram is a more detailed type of tree diagram, one in which probabilities are listed on the branches of every stage.
Example 20
To drive to work, Mr Katehos passes through 3 sets of traffic lights. The probability of a red signal (including amber) on each light is 0.3. Construct a tree diagram showing all possible arrangements of red and green signals for the 3 sets of lights. Hence, calculate the probability that on his way to work Mr Katehos meets:
(a) all green lights (b) 1 red followed by 2 greens (c) 1 red and 2 greens in any order (d) at least 1 red light
1 2 3
Corridor Floor plan
Solution
The tree diagram lists all possible outcomes along with the probabilities at each stage.
(a) P(GGG) = 0.7 × 0.7 × 0.7 = 0.343
(b) P(RGG) = 0.3 × 0.7 × 0.7 = 0.147 Multiplying along the RGG branch
(c) P(1R, 2G)= P(RGG) + P(GRG) + P(GGR) Ticked on the diagram
= 0.147 + (0.7 × 0.3 × 0.7) + (0.7 × 0.7 × 0.3) = 0.441
(d) Every outcome except one (GGG) has at least 1 red light, so we can use the complementary event rule.
P(at least 1 red)= 1 − P(no reds) = 1 − P(GGG)
= 1 − 0.343 From (a)
= 0.657
A probability tree diagram illustrates all of the possible outcomes of a multistage event, including their probabilities. The tree diagram above shows a 3-stage event.
Example 21
A student council has 8 Year 10 students, 6 Year 11 students and 4 Year 12 students. Two students are selected at random from the council to represent the school at the Lord Mayor’s lunch. Construct a probability tree diagram to show all possible selections and use it to calculate the probability that:
(a) both representatives are from Year 10
(b) there is one representative from each of Years 10 and 11 (c) at least one of the representatives is from Year 12 (d) each representative is from a different Year group
R
G
R
G
R
G
R
G
RRR
RRG
RGR
RGG ✔
GRR
GRG ✔
GGR ✔
GGG R
G
R
G
2nd signal 3rd signal Outcomes 1st signal
0.3
0.7 0.3
0.7 0.3
0.7 0.3
0.7 0.3
0.7
0.3
0.7 R
G 0.3
0.7
Multiplying along the branches of the GGG possibility
In a probability tree diagram:
n To calculate the probability of a particular outcome, multiply the probabilities listed on the branches of each stage of the outcome.
n To calculate the probability of an event with two or more outcomes, add their calculated probabilities together.
Solution
Total number of students = 8 + 6 + 4 = 18. The tree diagram shows all possible selections.
(a) P(both Y10) = × Multiplying along the Y10–Y10 branch
=
(b) P(Y10 and Y11)= P(Y10–Y11) + P(Y11–Y10) Marked by • on the diagram
=
=
(c) P(at least one Y12)=P(Y10–Y12) + P(Y11–Y12) + P(Y12–Y10) + P(Y12–Y11)
+ P(Y12–Y12) Marked by * on the diagram
=
=
(d) There are 6 outcomes involving representatives from different Year groups. However, there are only 3 outcomes not involving this, so it is more convenient to use the complementary event rule here.
P(different Years) = 1 − P(same Years)
= 1 − [P(Y10–Y10) + P(Y11–Y11) + P(Y12–Y12)] = 1 −
= 1 −
=
7 Year 10 students
Y10 Y11 Y12 Y10 Y11 Y12 Y10 Y11 Y12 Y10–Y10
Y10–Y11 •
Y10–Y12 *
Y11–Y10 •
Y11–Y11
Y11–Y12 *
Y12–Y10 *
Y12–Y11 *
Y12–Y12 *
Y10
Y11
Y12
1st rep 2nd rep Outcomes
8 17 4 17 5 17 8 17 3 17 6 17 7 17 4 17 6 17 8 18 4 18 6 18
left out of 17 students
8 18 --- 7 17 ---28 153 ---8 18 --- 6 17 ---×
6
18 --- 8 17 ---× + 16 51 ---8 18 --- 4 17 ---×
6
18 --- 4 17 ---×
4
18 --- 8 17 ---×
4
18 --- 6 17 ---×
4
18 --- 3 17 ---× + + + + 62 153 ---8 18 --- 7 17 ---×
6
18 --- 5 17 ---×
4
---1. A fruit bowl contains 3 oranges and 4 apples. Jim selects 2 pieces of fruit at random from the bowl. Use a tree diagram to determine the probability that Jim selects:
(a) 2 oranges (b) 2 apples (c) an apple and an orange
2. There are 10 batteries in a box, of which 2 are flat. Daniel takes 2 batteries out of the box at random. Calculate the probability that:
(a) both batteries are flat (b) only 1 of the batteries is flat
3. Over a long weekend (Saturday to Monday), the probability of rain on any particular day is 0.2.
(a) A tree diagram that illustrates this situation has been started below. Copy and complete the diagram.
(b) Calculate the percentage probability that over the long weekend there is: (i) exactly 1 rainy day (ii) no rainy days (iii) at least 1 rainy day
4. Tan buys 2 tickets in a 50-ticket raffle. There are two prizes. What is the probability that Tan wins:
(a) 1st prize? (b) 1st and 2nd prizes?
(c) no prize? (d) at least 1 prize?
5. Two cards are drawn at random from the 2 black kings and 2 black queens of a normal deck of cards. Calculate the probability that the cards drawn are:
(a) 2 kings (b) 2 queens
(c) a king and a queen of the same suit (d) a king and a queen of different suits
6. A die is rolled 3 times. What is the probability that 6 does not come up in any roll?
7. A tennis player gets a second serve only if his first serve does not go in. Patrick’s first serve has a 0.78 probability of going in and his second serve has a 0.94 probability of going in.
(a) Copy and complete the tree diagram.
(b) A double fault occurs when both the first and second serves do not go in. What is the probability that Patrick serves a double fault?
(c) What is the probability that one of Patrick’s serves goes in?
8. A committee of 4 women and 3 men need to select a chairperson and a secretary. If every person is equally likely to be chosen, what is the probability that:
(a) both positions are filled by women? (b) both positions are filled by men?
(c) the chairperson is female and the secretary is male?
9. A biased coin shows a tail 63% of the time. If it is tossed 3 times, calculate the probability (correct to 3 decimal places) that:
(a) a tail comes up every time (b) a tail comes up twice
Exercise 6-07:
Probability tree diagrams
R R
Saturday Sunday Monday
0.2
~
In
Not in
10. Five cards are numbered 1, 2, 3, 4 and 5. Two are randomly drawn. Find the probability of drawing:
(a) 2 even numbers (b) 2 odd numbers
11. Two cards are drawn from the same 5 cards as in question 10, but the first card is returned to the pack before the second is drawn. Find the probability of drawing:
(a) 2 even numbers (b) 2 odd numbers
12. A box contains 5 music CDs and 2 computer CD-ROMs. Karen chooses 3 at random. (a) Use a tree diagram to show that there are 7 possible selections.
(b) Calculate the probability that Karen chooses:
(i) the 2 CD-ROMs (ii) 2 music CDs
(iii) at least 1 of each type of CD (iv) at least 1 music CD
13. Three people are selected at random. What is the probability that all of them were born in March? Express your answer as a percentage correct to 3 significant figures.
14. 8% of a population is known to have a certain virus which can be detected by a medical test. However, this test is only 90% effective, meaning it gives a correct reading only 90% of the time. If Sam is tested for the virus, what is the probability that:
(a) he has the virus but it is not detected? (b) he has the virus and it is detected?
(c) he does not have the virus but it is falsely detected?
15. The probability that Christine, a hockey player, can score a goal is p. She takes 3 shots at the goal. Write an expression for the probability that Christine:
(a) scores 3 goals (b) scores no goals
(c) scores, misses and scores (in that order) (d) scores 2 goals
EXPECTATION
Example 22
In a Lotto draw, 6 numbers are selected from 1 to 44.
(a) What is the probability that Carlos’s lucky number, 34, is selected in a Lotto draw? (b) There are 104 Lotto draws in a year. How many times can Carlos expect his number to
be selected over the year?
Solution
(a) P(34) = = 6 chances of selecting 34
(b) Expected times = 104 ×
= 14 ≈ 14
Carlos can expect 34 to come up about 14 times over the year. 6
44 --- 3
22
---3 22
---2 11
---If the probability of an event E is p and the experiment is conducted n times, the expected number of times E will occur is
Example 23
A pair of coins are tossed 300 times. How often would you expect 2 heads to come up?
Solution
P(HH) = =
Over 300 trials, expected number of HH outcomes = 300 × = 75
Financial expectation
The financial expectation of a game of chance is the expected or average amount of money returned on each game.
Example 24
Samantha plays a game involving the tossing of 2 coins. Each game costs 40c to play. She wins $5 if both coins show a head, $1 for a head and a tail, but loses $6 if both are tails. (a) What is Samantha’s final expectation from the game?
(b) On average, will she make a profit or a loss?
Solution
(a) Using the tree diagram from Example 23
P(HH) = , P(HT or TH) = = , P(TT) =
Financial expectation= $5 × P(HH) + $1 × P(HT or TH) + (−$6) × P(TT) = $5 + $1 − $6
= $0.25
(b) Since the game costs 40c a play, and the expected return is 25c, Samantha can expect to lose an average of 40c − 25c = 15c per game.
Note: If the game cost less than 25c a play, Samantha would make a profit. If the game cost
exactly 25c, she would ‘break even’ in the long run and the game would be considered ‘fair’.
Example 25
Graham rolls a pair of dice in a game of chance that costs $1 per bet. The table lists the financial outcome for each event.
(a) Calculate the financial expectation for this game. (b) Is this game fair? Justify your
answer. H T H T HH HT TH TT H T
1st coin 2nd coin Outcomes
1 2 1 2 1 2 1 2 1 2 1 2 1 2 --- 1 2 ---× 1_{4}
---1 4
---Financial expectation is calculated by multiplying every possible financial outcome by its probability and adding the results together.
1 4 --- 2 4 --- 1 2 --- 1 4 ---1 4 ---
1
2 ---
1
4 ---
Event Financial outcome
Doubles (both dice the same) Win $2
Sum of 7 Win $3
Odd sum (except 7) Win $1 (money back)
Solution
(a) 36 possible outcomes
P(doubles)= = P(sum of 7)= = P(odd sum except 7)= = P(even sum except doubles)= =
Financial expectation= $2 + $3 + $1 + −$1 ≈ $0.83
(b) The game is not fair because the financial expectation ($0.83) is less than the cost of the game ($1). On average, you would be losing $0.17 per game. (However, most gambling games are like this so that the operator can make a profit.)
Is it fair that games of chance are arranged so that the casino or betting organisation always wins in the long run? Examine this issue from different points of view.
1. The probability of a new light globe not working is 0.036. If 5821 light globes are produced by a factory in one day, how many of them can be expected to be not working?
2. On a TV game show, a contestant selects from one of 9 panels which is turned around to reveal a prize. The panels contain 3 cash prizes, 2 holidays, 1 car and 3 other prizes. How often should a holiday be won over 260 shows?
3. A die is rolled 140 times. How many times should a 2 or a 5 come up?
4. Colin plays a game that costs $1 at a time. He tosses two coins and wins $2 for 2 heads, $1 for 2 tails and nothing otherwise.
(a) Calculate his financial expectation for each game. (b) Is this a fair game?
2nd die
+ 1 2 3 4 5 6
1 2 3 4 5 6 7
1st die
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
6 36 --- 1
6
---6 36 --- 1
6
---12 36 --- 1
3
---12 36 --- 1
3
---1 6 ---
1
6 ---
1
3 ---
1
3 ---
Think:
Is it fair that a casino always wins in the long run?
Exercise 6-08:
Expectation
1
7
2 3
4
5. One card is randomly selected from a normal deck of playing cards. If this experiment is run 40 times, how many times should:
(a) a queen be selected? (b) a club be selected?
(c) an odd-numbered card be selected?
6. Sonya pays 20c to roll a die. If she rolls a 6 she wins $1.05, but if she rolls an odd number she loses 5c.
(a) What is her financial expectation?
(b) Is this a fair game? If not, how much should the game cost to play so that it is fair?
7. The probability of a 20-year-old person dying this year is 11 in 10 000. (a) Express this probability as a decimal (i.e. a mortality rate).
(b) If the number of 20-year-olds in Australia is 1 861 590, how many of them are expected to die this year?
8. Grandad bet $5 on each of these 5 horses in a horserace.
(a) Calculate his financial expectation from this bet. (b) On average, will Grandad make a gain or a loss?
9. The probabilities of winning cash prizes on an instant lottery ticket are listed.
(a) What is the financial expectation from 1 ticket?
(b) If Kelly purchases 200 tickets at $2 each: (i) how much does she spend?
(ii) how much can she expect to collect?
10. In OzLotto, 6 numbers are chosen from 1 to 45. (a) How many different selections are possible?
(b) Hence, what is the probability of winning first prize with a single entry?
(c) One week, OzLotto sold 18 million entries for a draw with a first prize of $3 million. (i) How much money did OzLotto make if each entry costs 25c?
(ii) How many first-prize winners should OzLotto expect? (iii) What is each first-prize winner’s share?
11. The following game costs $1 to play. You reach into a bag containing 16 20-cent coins, 22 $1 coins and 12 $2 coins and take out 1 coin. You get to keep that coin.
(a) Calculate the financial expectation for this game. (b) Is this a fair game?
Horse Probability of win Payout from $5
Likely Lad 0.32 $11
Just a Minute 0.07 $68
Everyone’s a Critic 0.11 $42
Sauerkraut 0.01 $480
Solar Eclipse 0.05 $96
Prize Probability
$2 0.17
$5 0.08
$50 0.008
$100 0.0047
$500 0.0001
12. You are given two options:
A: a 100% chance of receiving $200 each week
B: a 70% chance of receiving $500 and a 30% chance of losing $200 each week In the long run, which is the better option?
13. A test has 3 sections. Section 1 has 20 multiple choice questions in which students choose 1 correct answer from 4 options (A, B, C, D). Section 2 has 10 true–false questions in which students choose either True or False. Section 3 has 10 multiple choice questions in which students choose 1 correct answer from 3 options (A, B, C). If every question in this test is randomly answered, what is the expected number of correct answers?
14. Out of 160 families that have 3 children, how many would you expect to have: (a) boy–girl–boy in that order?
(b) 2 boys and 1 girl? (c) at least 1 girl?
15. Two special dice are rolled together. The first die has two 6s but no 5 while the second die has two 1s but no 2. The sum of the dice is calculated.
(a) Copy and complete this table to show all possibilities.
(b) How many times should a sum of 9 come up if this pair of dice are rolled 80 times? (c) Ronnie wins $15 every time she rolls a sum of 5. If each roll costs her $2, how much
can she expect to win or lose in the long run?
16. The Getta Life insurance company has only 5 customers, whose details are listed below. The mortality rate is the chance of a person with that age dying this year. The size of the policy is how much the company must pay the person’s family if the person dies.
(a) If each person pays a premium of $250 each year, how much does the insurance company receive?
(b) What is the expected amount the company will pay out this year?
2nd die
+ 1 1 3 4 5 6
1 2 2 4 5 6 7
1st die
2
3
4
6
6
Name Age Probability Size of policy
Q. Jumper 35 0.0012 $50 000
T. Baggs 29 0.0004 $35 000
G. Whiz 51 0.0028 $42 000
L. Liphant 64 0.0129 $28 000
17. If this wheel is spun 60 times, how often can you expect: (a) the holiday to come up?
(b) the CDs to come up? (c) the dinner not to come up?
18. The Bulldogs have a 0.7 chance of winning their home games and a 0.41 chance of winning their away games. If they play 12 home games and 13 away games over a season, how many games should they win?
CDs
Holiday
Dinner
Computer
Cash
Just for the record
M
ORTALITY RATESOne important application of probability is insurance. Insurance companies rely on statistical data to calculate the probabilities of car accidents, injuries, burglaries, health problems and deaths. When a person takes out an insurance policy, he or she pays a regular amount (called a premium) to insure against an event occurring. If the event occurs, then the insurance company pays the person a large sum of money.
Insurance companies use mortality rates to calculate premiums for life insurance policies. With a life insurance policy, if a person dies the family receives the payout. Mortality rates are the probabilities that a person of a particular age will die that year. The table below shows the mortality rates of Australians aged 0 to 99 (1998 figures).
Source: Australian Bureau of Statistics, 1998.
1. How do you think these figures were calculated?
2. What patterns do you notice in the mortality rates as people get older? Why?
3. Why do you think babies have a higher mortality rate?
4. Why do you think there is such a difference between male and female mortality rates?
5. These mortality rates are much lower than those in 1988, and this year’s mortality rates would be lower than t