ChapTEr ConTEnTS
11a Introduction to experimental probability 11B Calculating probabilities
11C Tree diagrams and lattice diagrams 11d The Addition Law of Probabilities 11E Karnaugh maps and probability tables 11F Conditional probability
11G Transition matrices and Markov chains 11h Independent events 11i Simulation diGiTal doC doc-9801 10 Quick Questions
ChapTEr 11
Introductory probability
11a
introduction to experimental
probability
Tossing a fair coin or rolling a standard 6-sided die will result in a range of outcomes. The coin can land Heads or Tails, and the number appearing uppermost on the die will be one of the numbers 1, 2, 3, 4, 5 or 6. Probability involves assigning a numerical value to the likelihood of such events occurring.
In this respect, certain events will clearly be more probable than others; for example, getting only 1 of the required 6 numbers in Tattslotto is more likely than obtaining all 6 winning numbers.
A numerical value for the probability of an event can be established in a number of ways. It can be based on results arising from experiments; alternatively, a reasoned estimate of the likelihood of the event can be provided on the strength of personal experience and knowledge (the subjective
probability). A third way is to consider the ‘symmetry’ of the situation where the activity has
equiprobable or equally likely outcomes. For example, if we toss a coin 50 times and note how many times it lands ‘Heads’ (a Head facing up), we may conclude (based on the experiment) that the probability of a coin landing Heads up is half. We may also reason that a tossed coin has two equally likely outcomes (a Head and a Tail), of which Heads is one possibility, so there is 1 chance in 2, or 12, of a Head. However, deciding what the chances are of a runner winning her race will be subjective and dependent on considerations such as the runner’s past performances, her current state of fi tness and the abilities of the other competitors.
random outcome experiments
What is the probability of a fair coin landing Tails? For a single trial of this experiment (one toss of the coin), we know the coin will land either Heads or Tails, but we cannot be sure the toss will produce a
favourable outcome (that is, a Tail). The result is a random outcome. (Closing one’s eyes and taking out a marble from a box containing different coloured marbles, or shuffl ing a pack of playing cards and choosing the topmost card, are also activities that produce random outcomes.)
For our example of the coin, if many trials are conducted we will observe that the ratio number of Tails
total number of trials, which is the experimental probability for the favourable outcomes, converges (‘gets closer’) to a particular value. This particular value is known as the long-run proportion.
diGiTal doCS doc-9802
long run proportion doc-9803
one die doc-9804 Two dice
ConTEnTS
11a
introduction to experimental
probability 475
Exercise 11a
introduction to experimental probability
477
11B
Calculating probabilities 478
Exercise 11B
Calculating probabilities 481
11C
Tree diagrams and lattice diagrams
483
Exercise 11C
Tree diagrams and lattice diagrams
485
11d
The addition law of probabilities
487
Exercise 11d
The addition law of probabilities 490
11E
karnaugh maps and probability
tables 492
Exercise 11E
karnaugh maps and probability tables
Similarly, we observe that the ratio number of Heads
total number of trials, the experimental probability for a Head, converges to a particular value.
For a coin tossed many times the long-run proportion of a Head is 0.5 and the long-run proportion of a Tail is 0.5.
Experimental probability and expected number
of outcomes
In general, the experimental probability is given by:
experimental probability = number of favourable outcomes observed total number of trials
The number of times an outcome of an activity is expected to occur is given by: expected number of favourable outcomes
= experimental probability (long-run proportion) × number of trials
WorkEd ExamplE 1
A 6-sided die (not necessarily a fair one) was rolled 12 times and the number showing uppermost was noted each time. The numbers uppermost on the die were:
2, 4, 1, 1, 5, 6, 4, 3, 4, 5, 6, 1. Estimate the probability of rolling a 5 with this die.
Think WriTE
1 There are 2 favourable outcomes.
2 There are 12 outcomes altogether.
3 Use the formula: experimental probability = number of favourable outcomes observed
total number of trials
Experimental probability = 2 12
= 1 6
WorkEd ExamplE 2
A fair 6-sided die is rolled 48 times. How many times is an even number expected to show uppermost?
Think WriTE
1 There are 6 equally likely outcomes for the roll of the die and 3 favourable outcomes corresponding to an even number.
Experimental probability of an even number = 3
6
= 1 2 2 There are 48 trials. Use the formula
expected number of favourable outcomes = experimental probability × number of trials.
Expected number of even numbers = 1
2 × 48
= 24
WorkEd ExamplE 3
Inside a bag are 18 marbles, some white and the rest green. One marble is taken out without looking, its colour is noted and the marble put back inside the bag. When this is done 30 times it is found that a green marble was taken out 5 times. Estimate how many marbles of each colour are in the bag.
Think WriTE
1 A green marble was taken out 5 times and a white marble 25 times. Work out the experimental probabilities.
Experimental probability of green marble = 5 30
= 1 6
Experimental probability of white marble = 25 30
= 5 6 2 Calculate the expected number of each colour
marble. Use the formula
expected number of favourable outcomes = experimental probability × number of trials.
Expected number of green marbles = 1 6 × 18
= 3 Expected number of white marbles = 5
6× 18
= 15 Estimated number of each type of marble: 3 green, 15 white
Exercise 11a
introduction to experimental probability
1 WE 1 A coin was tossed 10 times and the outcomes noted as H, T, H, H, T, T, T, T, H, T, where H is a Head and T is a Tail. Find the experimental probability of a Tail.
2 Twenty letters were chosen at random from the alphabet and recorded as either consonant, c, or vowel, v. The results were c, c, v, c, v, c, c, c, c, v, c, v, v, c, c, c, c, v, c, c. Calculate the experimental probability of choosing a consonant.
3 A biased coin is tossed 50 times. The results were 33 Tails and 17 Heads.
a What is the experimental probability of tossing a Tail with this coin?
b What is the experimental probability of tossing a Head with this coin?
4 WE2 A die is tossed 96 times. How many times is an odd number expected to appear uppermost on the die?
5 A coin is tossed 500 times. What is the expected number of Heads?
6 A die is rolled 300 times. How many odd numbers or the number 2 are expected to turn up?
7 mC A die is tossed 102 times. The number of times a number between 1 and 3 inclusive is expected to appear uppermost on the die is:
a 51 B 34 C 20 d 64 E 68
8 mC A box contains 2 blue beads, 3 green beads and 1 yellow bead. One bead is taken out, its colour
is noted and it is put back in the box. This is repeated 246 times. The number of times a bead that is not yellow is expected to be taken out of the box is:
a 41 B 82 C 205 d 123 E 164
9 WE3 Inside a box are 42 plastic shapes. Some of the shapes are squares and the remainder are circles. One shape is taken out at random, its shape is noted and it is put back in the box. After this is repeated 84 times it is found that a square was taken out 36 times. Estimate how many squares and how many circles are in the bag.
10 A closed box with a hole in one corner contains coloured marbles: 4 are red, 2 are blue, 3 are white and 1 is green. The box is shaken and 1 marble falls out. Its colour is recorded and it is placed back in the box. This is done 200 times.
a How many times is a red or blue marble expected to fall out of the box after 200 trials?
b How many times is a marble that has a colour other than white expected to fall out of the box after 200 trials?
11 mC A moneybox contains 128 coins. There are 5-cent and 10-cent coins. The box is shaken, a coin falls out, the value of the coin is noted and it is placed back inside the box. After this is repeated 96 times it is noted that a 5-cent coin fell out 60 times. The estimated number of 10-cent coins in the moneybox is:
12 During a period of one week 190 people telephoned Hot-Shot Electrics with enquiries. During the same period Zap Inc received 305 enquiries. Based on this information, how many enquiries did Hot-Shot and Zap Inc each expect during a week where the total number of phone calls made to the two businesses was 650 (to the nearest call)?
13 The probability that the Last Legs football team can win a match is 2
7. If the team is to play 35 matches during the season, how
many wins should it expect?
14 In a 9-game chess tournament, Adam won 6 games, lost 2 games and drew 1 game.
a Based on this information, if Adam is to play 108 games next year, how many games should he expect to:
i win? ii lose? iii draw?
b Based on the fact that Adam won 81 of the 108 games, how many games does he expect to lose or to draw in a tournament comprising 16 games?
15 mC Inside a bag are red, blue and black marbles. Sally takes out one marble, notes its colour and puts it back in the bag. When she has taken out a marble 360 times, she finds that a red marble was taken out 140 times and a black marble 200 times. If Sally takes out a marble 270 times, the number of blue marbles expected is:
a 15 B 90 C 105 d 150 E 125
16 A post-office has two letterboxes, Domestic and Overseas. Letters to be delivered within Australia are placed in the Domestic box, and letters intended for overseas destinations are deposited in the Overseas box. During the month of January there were 980 Domestic and 310 Overseas letters handled.
a Estimate the probability of a particular letter having an Australian destination.
b Estimate the probability of a particular letter having an overseas destination.
c During February there were 1580 letters posted in total. How many of these would you expect to have been delivered: i within Australia? ii overseas?
11B
Calculating probabilities
Many of the methods arising from a study of probability can be investigated by using set theory. A review of the basic work on sets is provided in your eBookPLUS.
This section describes how to calculate exact theoretical probabilities rather than use experimental results to estimate probabilities. We know that the theoretical probability of a fair coin coming up Heads is exactly 12. We must remember that this does not mean that exactly half the tosses of the coin will be Heads, but rather that the long-run proportion of Heads will approach 12 as the number of tosses becomes very large.
If n is the number of trials, then as n → ∞ (→ means ‘approaches’, or ‘gets closer to’): the proportion of successes → the theoretical probability of a ‘success’.
Before describing how to calculate theoretical probability (and avoid the need to perform a large number of trials), we need to discuss the ideas of event space and favourable outcome in more detail.
Event space
The event space (or sample space) consists of all possible outcomes of an experiment. The event space is the universal set and is denoted by ξ.
WorkEd ExamplE 4
A 6-sided die is rolled. List the elements of the event space and state the cardinal number.
Think WriTE
1 List the elements of the event space. ξ = {1, 2, 3, 4, 5, 6}
2 Count the number of elements in the event space. n(ξ) = 6
diGiTal doC doc-9805 WorkSHEET 11.1 diGiTal doC doc-9810 Extension Sets and Venn diagrams
WorkEd ExamplE 5
A coin and a die are tossed.
a List the elements of the event space.
b List the elements of the event E = ‘Head and a number greater than 4’.
Think WriTE
a What are the different outcomes using a
coin and a die together? a
Let H be Head, T be Tail and H4 mean ‘Head on the coin and a 4 on the die’.
Then ξ = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}.
b List all the possible ways of obtaining event
E = ‘Head and a number greater than 4’. bEvent E consists of 2 sample points: E = {H5, H6}.
probabilities
The game of ‘Zilch’ involves tossing a fair 6-sided die and scoring points for rolling a 6 or a 1. The events ‘rolling a 6’ and ‘rolling a 1’ are called ‘favourable outcomes’.
The total number of outcomes is 6 (a result of 1, 2, 3, 4, 5 or 6 could be rolled). Each outcome is equally likely for a fair die.
Intuition may lead us to assert that the probability of scoring by rolling a die in a game of Zilch is 2
6= 1 3.
More formally, for equally likely outcomes:
probability of a favourable outcome = number of favourable outcomes total number of possible outcome or
Pr(favourable outcome) = number of favourable outcomes total number of possible outcome So in our Zilch example,
Pr(scoring) = number of favourable outcomes total number of possible outcome Pr(scoring) = 2
6
= 1
3 as before.
There are other, equivalent expressions for calculating probability, including Pr(E) = number of favourable outcomes in E
total number of possible outcomes Or, using set notation,
Pr(E) =n(E ) n(ξ )
where Pr(E ) is the probability of event E, n(E ) is the cardinal number of event E and n(ξ ) is the cardinal number of the event space. The above Zilch example may be illustrated as follows, where E = getting a 6 or a 1.
Notice that Pr(E ′) + Pr(E ′) = 2 6 +
4 6
= 1. In general, if E and E ′ are complementary events,
Pr(E) + Pr(E′) = 1 and Pr(E′) = 1 − Pr(E) ξ E 6 2 3 5 4 1
WorkEd ExamplE 6
A number is randomly chosen from the first 12 positive integers. Find the probability of: a choosing the number 8
b choosing any number except 8.
Think WriTE
a 1 Pr(favourable outcome)
= number of favourable outcomes total number of possible outcomes
a
2 There is one favourable outcome (choosing an 8) and 12 possible outcomes.
Pr(8) = 1 12
bWe require the complementary probability.
Pr(E ′ ) = 1 − Pr(E ). bPr(not 8) = 1 −
1 12
= 11 12
range of probabilities
If there is no favourable outcome for event E, then n(E ) = 0, so: Pr(E) = n(E )
n(ξ ) = 0
n(ξ ) = 0
We interpret this to mean that impossible events have a probability of zero. If every outcome in the event space for E is a favourable outcome, then
n(E ) = n(ξ) and Pr(E) = n(E ) n(ξ ) = n(ξ ) n(ξ ) = 1.
We interpret this to mean that events certain to happen have a probability of 1. Thus the range of values for the probability of an event is given by 0 ≤ Pr(E ) ≤ 1. The probability line below illustrates the range of probabilities and the likelihood of the event occurring.
Impossible Unlikely Equally
likely Likely Certain
1– 2 1– 4 3–4 0 1 WorkEd ExamplE 7
A fair cubic die with faces numbered 1, 3, 4, 6, 8, 10 is rolled. Determine the probability that the number appearing uppermost will be:
a even b odd c less than 1 d greater than or equal to 1.
Think WriTE
a 1 There are 6 possible outcomes when rolling the die. a n(ξ ) = 6
2 Four of the outcomes correspond to an even
number. Pr(even number) =
4 6
= 2 3
bTwo of the outcomes correspond to an odd number. Also 1 − 2
3= 1 3 since
Pr(odd number) + Pr(even number) = 1
b Pr(odd number) = 26
= 1 3
cNone of the outcomes correspond to a number less than 1.
cPr(number is less than 1) = 0 6
= 0
dAll 6 outcomes correspond to a number greater than or equal to 1.
dPr(number is greater than or equal to 1) = 6 6
= 1 In worked example 7, notice that if A is the event ‘even number’ then the complement (A′) of A is the event ‘odd number’ and
Pr(A) + Pr(A′) = 4 6 +
2 6
= 1.
Similarly, if B is the event ‘a number less than 1’ then B′ is the event ‘a number greater than or equal to 1’ so that Pr(B) + Pr(B′) = 0 6 + 6 6 = 1 WorkEd ExamplE 8
One letter is randomly selected from the letters in the sentence LITTLE MISS MUFFETT. Calculate the probability that the letter is:
a a vowel
b a consonant other than a T c a consonant.
Think WriTE
a 1 Pr(favourable outcome)
= number of favourable outcomes total number of possible outcomes
a
2 A vowel is a favourable outcome. There are 17 possible outcomes (letters), of which 5 are vowels.
3 Substitute this information into the probability
formula. Pr(vowel) =
5 17
b 1 There are 8 consonants other than T. b
2 Use the probability formula. Pr(consonant other than T) = 8 17
c 1 There are 12 consonants. c
2 Write the probability. Pr(consonant) = 12 17
Exercise 11B
Calculating probabilities
1 WE4 A spinner is divided into 4 equal sections as shown at right. For one spin:
a list the elements in the event space
b state the cardinal number of the event space.
Blue
Green Yellow
2 A numberplate is made up of 3 letters followed by 3 numbers. What is the event space for the first position on the numberplate?
3 A card is chosen from a pack of 52 playing cards. What is the cardinal number of the event space?
4 WE5 A card is chosen from a pack of 52 playing cards and its suit noted, then a coin is tossed.
a List the elements in the event space.
b List the elements in the event S = ‘a spade is chosen’.
5 A coin is tossed twice. List the elements in the event space.
6 A student is chosen at random from a class of 12 girls and 14 boys, then a chocolate bar is chosen from a bag containing a Time Out, a Mars Bar and a Violet Crumble.
a List the elements in the event space.
b List the elements in the event M = ‘a Mars Bar is chosen’.
7 WE6 One player is chosen at random from the senior netball team to be the captain. If there are 7 players in the team, what is the probability the person who plays goal attack is:
a chosen? b not chosen?
8 One Year 11 student must be chosen to represent the year level at a staff meeting. If all 81 girls’ and 73 boys’ names are put into a container and one name is chosen at random, find the probability that:
a a Year 11 student is chosen
b any particular Year 11 student is chosen
c a boy is chosen.
9 One card is chosen from a pack of 52 playing cards. What is the probability that the card is:
a a queen? b a heart?
c a picture card (J, Q, K)? d not a picture card?
e red or black?
10 Four hundred thousand tickets are sold in a raffle. The winner of the raffle will toss the coin at the AFL grand final. If you bought 10 tickets, what is the probability that you will win?
11 WE7 A standard die is rolled. What is the probability of rolling:
a an even number?
b a 5?
c a number from 2 to 4 inclusive?
d a number less than 7?
12 A bag has 20 marbles numbered 1, 2, 3, . . . , 20. One marble is randomly drawn. Find the probability that the number on the marble is:
a even b greater than 4 c a multiple of 4 d not even.
13 WE8 One letter is randomly selected from the letters of the palindrome ‘Madam, I’m Adam’. Calculate the probability that the letter is:
a a vowel b a consonant other than aD.
14 What is the probability of randomly choosing a consonant other than P from the letters of the palindrome ‘A man, a plan, a canal, Panama’?
15 One letter is randomly selected from the words Mathematical Methods. What is the probability of randomly selecting:
a the letter m?
b a vowel?
c a consonant?
d a letter from the first half of the alphabet?
16 A lolly shop has 85 different types of lollies including Smarties in clear plastic containers. Forty of the lollies contain chocolate. If one container is chosen at random, what is the probability it contains:
a a lolly containing chocolate?
b Smarties?
11C
Tree diagrams and lattice diagrams
Tree diagrams
A useful way of representing all possible outcomes for sequential activities is by means of a tree diagram. A tree diagram consists of paths formed from branches. Each sample point (possible outcome) corresponds to a unique path that is found by following the branches. For example, a tree diagram could be drawn to show all possible outcomes when a coin is tossed twice.
The fi rst set of branches shows the possible outcome of the fi rst activity, in this case tossing the coin the fi rst time.
The second set of branches is then joined onto the ends of the fi rst set to show all outcomes of both tosses of the coin.
Note that the outcomes are written at the end of each path through the tree diagram.
The cardinal number of the sample space is the total number of end branches at the end of each path. If all outcomes are equally likely, the probability can then be determined as before by using Pr(E ) = n(E )
n(ξ ).
WorkEd ExamplE 9
A card is chosen from a pack of 52 playing cards and its suit noted; then it is returned to the pack before another card is chosen.
a Draw a tree diagram showing all possible suit outcomes. b Calculate the probability of choosing:
i two hearts ii a diamond then a spade iii a heart and a club.
Think WriTE/draW
a Draw a tree diagram. S — spade
H — heart D — diamond C — club
a 1st card 2nd card Outcome S H D C SS S H D C S H D C S H D C S H D C SH SD SC HS HH HD HC DS DH DD DC CS CH CD CC
b iUse the probability formula with one favourable outcome (heart, heart) out of 16 possible outcomes.
b iPr(HH) = 1 16
iiUse the probability formula with one favourable outcome (diamond, spade) out of 16 possible outcomes.
iiPr(DS) = 1 16
iiiUse the probability formula with two favourable outcomes (heart then club or club then heart) out of 16 possible outcomes.
iiiPr(HC or CH) = 162 = 1 8 1st coin H T 1st coin 2nd coin H T H T H T HH HT TH TT Outcome TUTorial eles-1448 Worked example 9
WorkEd ExamplE 10
Two letters are selected from the word BIRD. a Draw a tree diagram to illustrate the event space.
b What is the probability that the second letter is a vowel or that the first letter is D?
Think WriTE/draW
a 1 There are 4 letters to choose from as the first
letter of the pair of letters. a 1st letter 2nd letter Outcomes
B R D B I D B I R I R D I B R D BI BR BD IB IR ID RB RI RD DB DI DR 2 For each letter chosen as the first letter, there
are 3 letters remaining to choose from.
bThere are 5 favourable outcomes {BI, RI, DI, DB and DR} and 12 outcomes altogether. b
Pr(second letter is a vowel or first letter is D) = 5
12
lattice diagrams
When showing all possible outcomes of two activities such as ‘a die is rolled twice’, a tree diagram can become very large. An alternative method of showing all possible outcomes in this situation is a lattice diagram.
A lattice diagram is a graphical representation in which the axes show the possible outcomes of each activity. The ‘coordinates’ or points inside the graph show the possible outcomes from the combination of both activities, for example a total. These can be written as dots, as totals or by using a symbol for each outcome. Die 1 Die 2 1 2 3 4 5 6 1 2 3 4 5 6 Die 1 Die 2 1 2 3 4 5 6 1 2 3 4 5 6 2 3 4 5 6 7 3 4 5 6 7 8 4 5 6 7 8 9 5 6 7 8 9 10 6 7 8 9 10 11 7 8 9 10 11 12 Die Coin H T 1 2 3 4 5 6 H1 T1 H2 T2 H3 T3 H4 T4 H5 T5 H6 T6 WorkEd ExamplE 11
A die is rolled twice.
a Draw a lattice diagram to show all of the possible outcomes. b Find the probability of rolling a 2, then a 1.
Think WriTE/draW a The possible outcomes from each roll are
1, 2, 3, 4, 5, 6.
Put these numbers on each axis.
a 1 2 3 4 5 6 1 2 3 4 5 6
b The probability of obtaining a 2 on the fi rst die and a 1 on the second is shown by one outcome only. There are 36 total possible outcomes.
bPr(2, 1) = 1 36
c 1 The question asks for the probability of 2 numbers and the probability of a total of 7, so write totals on the diagram to show the possible outcomes of both events.
There are 6 ways of getting a total of 7 from a total of 36 possible outcomes.
{1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1} c Die 1 Die 2 1 2 3 4 5 6 1 2 3 4 5 6 2 3 4 5 6 7 3 4 5 6 7 8 4 5 6 7 8 9 5 6 7 8 9 10 6 7 8 9 10 11 7 8 9 10 11 12
2 Write the answer. Pr(total = 7) = 6 36
= 1 6
Exercise 11C
Tree diagrams and lattice diagrams
1 WE9 A psychic powers test kit contains 10 blue, 10 red and 10 green cards, each without any markings. In one particular test session, ‘Mental Mal’ selects a card, replaces it, and selects a card again.
a Draw a tree diagram showing the possible colour outcomes at each stage.
b Calculate the probability of Mal choosing:
i two blue cards
ii a red card, then a green card
iii a green and a red card.
2 A coin is tossed together with a disc that is red on one side and white on its other side.
a Show all possible outcomes on a tree diagram.
b Calculate the probability that the coin lands Tails and the disc lands red.
3 Two letters from the word CAT are chosen.
a Show all possible outcomes on a tree diagram.
b Calculate the probability that the letter A is chosen fi rst and the letter T is chosen second.
4 Two coins are tossed.
a Show all possible outcomes on a tree diagram.
b Find the probability that one head and one tail turned up.
5 The two spinners shown are spun and the colour on which each stops is noted. Find the probability that the spinners land on:
a red and green
b yellow and blue
c yellow and green.
diGiTal doC doc-9806 Stirling’s formula
Spinner 2 Spinner 1
6 mC A coin is tossed and a wheel that is coloured blue, white and yellow is spun.
The probability of getting Tails and the colour yellow is:
a 56 B 16 C 45
d 12 E 34
7 A pentagonal solid whose faces are numbered 2, 4, 6, 8, 10 is rolled and a disc that is red on one side and blue on its other side is tossed. Draw a tree diagram and calculate the probability that a number greater than 4 is rolled and the colour showing uppermost on the disc is red.
8 WE 10 An integer from 2 to 3 inclusive is chosen from one hat and an integer from 4 to 6 inclusive is chosen from another hat.
Draw a tree diagram showing the possible outcomes and determine the probability of selecting:
a two even numbers
b two odd numbers
c two even numbers or two odd numbers.
9 Peter chooses to wear a jacket and tie from the available jackets and ties on his clothes rack, which is shown in the photo at right. Use the photo to draw a tree diagram showing the possible jacket and tie choices. Calculate the probability of choosing the darker brown jacket with the red and yellow tie.
10 Each of the smaller triangles formed by the intersection of the diagonals of a square is painted using either red, green or blue before covering each one with a low-sheen or full-gloss varnish. If the colour of each triangle is chosen at random, draw a suitable tree diagram and find the probability that the triangle is not coloured red or green and is covered with full-gloss varnish.
11 A coin is tossed three times.
a Show all possible outcomes on a tree diagram.
b Find the probability of getting Head, Tail, Tail.
c What is the probability of getting at least two Tails?
12 Johnny wishes to try all combinations of a supercone ice-cream that has three scoops of different flavours chosen from chocolate, vanilla, strawberry, lime and banana. The middle scoop must be chocolate.
If Johnny randomly chooses his supercone ice-cream, show all possible outcomes on a tree diagram.
13 Alan, Bjorn and Carl each toss a coin at the same time. Draw an appropriate tree diagram and use it to find the probability that Alan’s and Carl’s result will both be Tails.
14 A consonant is selected from each of the words MATHS IS FUN.
a Show the possible outcomes on a tree diagram.
b Find the probability that the letters H and S will appear in the selection.
15 Two coins are tossed and a die is rolled. One of the coins is double-headed. Find the probability that you get:
a two Heads and an even number
b a Head, a Tail and an odd number
c a Head, a Tail and a number less than 4.
16 Find the probability of obtaining an odd number and at least one Tail when a die and two coins are tossed.
17 mC Three coins are tossed once. The probability that at least one coin shows Heads is:
a 3 4 B 3 8 C 2 3 d 7 8 E 1 4 Chocolate
18 Four coins are tossed.
a Show all possible outcomes on a tree diagram.
b Find the probability of obtaining Head, Tail, Head, Tail in that order.
c Find the probability of obtaining two Heads and two Tails.
d Find the probability of obtaining at least two consecutive Tails.
19 WE 11 a Draw a lattice diagram to show all possible outcomes when two dice are rolled.
b Use the lattice diagram to find the probability that both the numbers appearing uppermost are odd numbers.
c Find the probability of getting a total of 9.
20 A die is rolled and a coin is tossed.
a Draw a lattice diagram to show all of the possible outcomes.
b Find the probability of obtaining a 3 and a Tail.
c Find the probability of obtaining an even number and a Head.
21 Two dice are rolled. Find the probability:
a of obtaining two 6s
b of rolling a 3 and a 4
c that the sum of the numbers appearing uppermost is less than 10
d that the first number is a 3 and the sum of the numbers appearing uppermost is less than 8
e of rolling two multiples of 2.
22 A die labelled with the letters T, O, M, A, T, O and a die numbered 3, 4, 5, 6, 7, 8 are rolled together. Determine the probability that the first die shows a vowel and the second die shows a number greater than 6.
23 A diner orders an entree, main course and dessert from a lunch menu that offers 3 different entrees, 2 different main courses and 2 different desserts. Show these choices on a tree diagram and find the probability that the diner orders a particular entree and main course.
11d
The addition law of probabilities
Recall from our review of set theory that:
n(A ∪ B) = n(A) + n(B) − n(A ∩ B) [1] We also know that:
Pr(A ∪ B) = n(A ∪ B)
n(ξ ) [2]
Substituting [1] into [2], we get:
Pr(A ∪ B) = n(A) + n(B) − n(A ∩ B)
n(ξ ) = n(A) n(ξ ) + n(B) n(ξ ) − n(A ∩ B) n(ξ ) So, Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B).
Since we may equate ∪ with OR and ∩ with AND, we can say: Pr(A or B) = Pr(A) + Pr(B) − Pr(A and B) This is known as the Addition Law of Probabilities.
mutually exclusive events
If A ∩ B = ϕ, then A and B are mutually exclusive. That is, events A and B cannot happen at the same time. If A and B are mutually exclusive, Pr(A ∩ B) = 0, the Addition Law becomes:
Pr(A ∪ B) = Pr(A) + Pr(B) or
WorkEd ExamplE 12
If A and B are events such that Pr(A) = 0.8, Pr(B) = 0.2 and Pr(A ∩ B) = 0.1, calculate Pr(A ∪ B).
Think WriTE
Substitute the values for Pr(A), Pr(B) and Pr(A ∩ B) in the Addition Law to fi nd Pr(A ∪ B).
Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B) Pr(A ∪ B) = 0.8 + 0.2 − 0.1
Pr(A ∪ B) = 0.9
WorkEd ExamplE 13
If A and B are events such that Pr(A ∪ B) = 0.55, Pr(A) = 0.2 and Pr(B) = 0.45, calculate
Pr(A ∩ B).
Think WriTE
1 Substitute the values for Pr(A ∪ B), Pr(A)
and Pr(B). Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B)0.55 = 0.2 + 0.45 − Pr(A ∩ B)
2 Rearrange the expression to fi nd Pr(A ∩ B). 0.55 = 0.65 − Pr(A ∩ B)
Pr(A ∩ B) = 0.1
WorkEd ExamplE 14
If Pr(A ∩ B) = 0.2 and Pr(A ∪ B) = 0.9, calculate Pr(A) and Pr(B) if events A and B are equally
likely to occur.
Think WriTE
1 Use the Addition Law. Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B)
2 If events A and B are equally likely to occur,
then Pr(A) = Pr(B). Let x represent Pr(A) and hence Pr(B).
3 Substitute the information into the
Addition Law and solve. 0.9 0.9 = x + x − 0.2= 2x − 0.2 1.1 = 2x so x = 0.55
Pr(A) = 0.55, Pr(B) = 0.55
WorkEd ExamplE 15
A box contains 16 marbles numbered 1, 2, 3, . . . , 16. One marble is randomly selected.
Let A be the event ‘the marble selected is a prime number greater than 3’ and let B be the event ‘the marble selected is an odd number’.
a Evaluate:
i Pr(A) ii Pr(B) iii Pr(A ∩ B) iv Pr(A ∪ B).
b Are A and B mutually exclusive events?
Think WriTE
a 1 Write down the elements of A, B,
A∩ B and A ∪ B. aAB= {5, 7, 11, 13} = {1, 3, 5, 7, 9, 11, 13, 15} A∩ B = {5, 7, 11, 13} A∪ B = {1, 3, 5, 7, 9, 11, 13, 15} 2 What is n(ξ)? n (ξ) = 16 TUTorial eles-1449 Worked example 15
3 Calculate the probability of A, B, A∩ B and A ∪ B. i Pr(A) = 4 16 = 1 4 ii Pr(B) = 8 16 = 1 2 iii Pr(A ∩ B) = 4 16 = 1 4 iv Pr(A ∪ B) = 8 16 = 1 2
b Since Pr(A ∩ B) ≠ 0, it follows that A and B are not mutually exclusive.
bFrom iii above, we see that Pr(A ∩ B) = 1 4.
As Pr(A ∩ B) ≠ 0, A and B are not mutually exclusive.
Note: The Addition Law could also be used to determine any one of Pr(A ∪ B), Pr(A), Pr(B) or Pr(A ∩ B) when the other three quantities are known.
For example, to find Pr(A ∪ B) we have:
Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B) = 4 16+ 8 16− 4 16 = 1 2
The Venn diagram below left may be adapted to show probabilities rather than outcomes and used to solve problems (below right).
ξ A B (A ∪ B)′ or A∩ B A∩ B′ A′ ∩ B A′ ∩ B′ ξ Pr(A ∪ B)' or Pr(A' ∪ B') A B Pr(A ∩ B) Pr(A ∩ B') Pr(A' ∩ B) WorkEd ExamplE 16
An 8-sided die (numbered from 1 to 8) is rolled once. Find the probability that the number appearing uppermost is: a an even number b an even number or a multiple of 3.
Think WriTE
a 1 Let E = even number = {2, 4, 6, 8}. The probability of getting an even number = n(E ) n(ξ ). aPr(E ) = n(E ) n(ξ ) 2 n(E) = 4, n(ξ ) = 8 Pr(E) = 4 8 = 1 2
b 1 M = multiple of 3 = {3, 6}. The probability of an even number or a multiple of 3 = Pr(E ∪ M ).
bPr(E ∪ M) = Pr(E) + Pr(M ) − Pr(E ∩ M )
2 Pr(E ) = 1 2, Pr(M ) = 2 8 = 1 4, E∩ M = {6} so Pr(E ∩ M ) = 1 8. Pr(E ∪ M ) = 1 2 + 1 4 − 1 8 = 5 8
WorkEd ExamplE 17
If Pr(A) = 0.6, Pr(B) = 0.45 and Pr(A ∪ B) = 0.7, show this information on a Venn diagram and
calculate Pr(A ∪ B)′.
Think WriTE/draW
1 Draw a 2-set Venn diagram with an overlapping region.
2 Calculate the probability of the overlap, Pr(A ∩ B),
using the Addition Law. Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B) 0.7 = 0.6 + 0.45 − Pr(A ∩ B) Pr(A ∩ B) = 1.05 − 0.7
= 0.35
3 Complete the Venn diagram using the available
information. ξ A B
(A ∪ B)'
0.25 0.35 0.1
4 Calculate Pr(A ∪ B)′. Pr(A ∪ B) + Pr(A ∪ B)′ = 1
Pr(A ∪ B)′ = 1 − Pr(A ∪ B) = 1 − 0.7 = 0.3
Exercise 11d
The addition law of probabilities
1 WE12 If Pr(A) = 0.4, Pr(B) = 0.5 and Pr(A ∩ B) = 0.2, what is Pr(A ∪ B)?
2 If Pr(A) = 0.65, Pr(B) = 0.25 and Pr(A ∩ B) = 0.22, what is Pr(A ∪ B)?
3 If A and B are mutually exclusive events and Pr(A) = 0.38, Pr(B) = 0.51, what is Pr(A ∪ B)?
4 WE13 If A and B are events such that Pr(A) = 0.4, Pr(B) = 0.5 and Pr(A ∪ B) = 0.6, calculate Pr(A ∩ B).
5 For events X, Y, if Pr(Y) = 0.44, Pr(X ∩ Y) = 0.16 and Pr(X ∪ Y) = 0.73, what is Pr(X)?
6 For events D and E, if Pr(D) = 0.76, Pr(D ∪ E) = 0.82 and Pr(D ∩ E) = 0.35, what is Pr(E)?
7 WE14 If Pr(A) = 2 × Pr(B), Pr(A ∩ B) = 0.23 and Pr(A ∪ B) = 0.94, determine the values of Pr(A) and Pr(B).
8 If Pr(A ∪ B) = 0.75, Pr(A) = 0.28 and Pr(B) = 0.47, what can be concluded about the relationship between A and B?
9 If Pr(A ∩ B) = Pr(A), what is the relationship between A and B?
10 WE15 A card is chosen at random from a pack of 52 playing cards. Let H be the event ‘choosing a heart’ and P be the event ‘choosing a picture card (J, Q, K)’.
a Evaluate:
i Pr(H)
ii Pr(P)
iii Pr(H ∩ P) iv Pr(H ∪ P).
b Are H and P mutually exclusive events?
11 A box of chocolates contains 12 with hard centres and 8 with soft centres. One chocolate is chosen at random. Let H be the event ‘choosing a hard centre’ and S be the event ‘choosing a soft centre’.
a Evaluate:
i Pr(H)
ii Pr(S)
iii Pr(H ∩ S)
iv Pr(H ∪ S).
12 From a group of 100 people, 25 said they drink tea, 40 said they drink coffee and 15 said they drink both beverages. If one member of the group is randomly chosen, what is the probability that the person:
a drinks only tea?
b drinks neither tea nor coffee?
c drinks tea and coffee?
d drinks tea or coffee?
13 WE16 A box contains 20 marbles numbered 1 to 20. Find the probability of obtaining:
a an even number
b a multiple of 3
c a multiple of 2 or 3.
14 Find the probability of an odd number or a multiple of 4 appearing uppermost when a die is rolled.
15 Find the probability that a number divisible by 4 or 5 is drawn from a ‘lucky dip’ containing the first 50 natural numbers.
16 From a standard pack of 52 playing cards, one card is randomly drawn. State the probability that the card is:
a a ten
b a diamond
c a king or a jack
d a diamond, a spade or the ace of hearts.
17 A mixed bag of lollies contains 8 peppermint twists, 10 red jelly beans, 10 caramels, 18 chocolates, 4 peppermint twirls, 5 yellow jelly beans and 25 toffees. If Tara randomly selects one lolly, what is the probability that it is:
a a peppermint or a jelly bean?
b not a toffee, a caramel or a jelly bean?
c a peppermint, given that the jelly beans are stuck together and cannot be selected?
18 Sarah is competing in a 400-metre race against 13 other runners. If each contestant has the same probability of winning, find the probability that Sarah:
a wins the race
b comes first or second
c finishes in the top four
d does not qualify for the final 5.
19 A moneybox containing eight $1 coins, five $2 coins, nine 50c coins and two 20c coins is shaken and one coin falls out. Assuming that each coin is equally likely to fall out, calculate the probability that the coin’s value is:
a between 10c and $2 (not including 10c or $2)
b not 50c
c $1 or $2
d less than $1.
20 Inside a dresser drawer are 4 ties, 10 socks, 4 handkerchiefs and 2 towels. If Tony randomly takes out one item, find the probability that it is:
a something to be worn
b not a towel and not a sock
c either a sock or not a sock
d either a towel or not a handkerchief.
21 A patron in a restaurant is presented with a fruit platter consisting of 6 whole apples, 8 slices of orange, 5 sliced pear pieces, 11 whole strawberries, 6 whole plums and 4 sliced apricot halves. The waiter accidentally trips and a piece of fruit falls off the platter. Assuming that each piece of fruit was equally likely to fall, state the probability that the fallen fruit is:
a not a plum and not an apricot
b not sliced
c sliced or is not a strawberry
d either a pear or an orange that has not been sliced.
22 The games Alotto, Blotto and Clotto involve guessing a number from 1 to 100 inclusive. To win Alotto the number guessed must be a multiple of 3. To win Blotto the number must be a multiple of 5 or a multiple of 8. To win Clotto the guessed number is to be between 10 and 20 or greater than 77.
23 WE17 If Pr(A) = 0.3, Pr(B) = 0.4 and Pr(A ∪ B) = 0.65, show this information on a Venn diagram and find Pr(A′ ∪ B).
24 Of 20 people interviewed, 7 stated that they use both a tram and a train to get to work, and 2 said they drive their own car. No other form of transport or combination of transport is
used. If 5 people travel only by train, find the probability that a person selected at random travels by tram only.
25 The unusual dartboard shown below consists of 10 concentric circles, with 1024 points given for a dart landing within the first (smallest) circle, 512 points for a hit within the area bounded by the first and second circle, 256 points if the dart lands within the area bounded by the second and third circles, and so on. The area bounded by any two consecutive circles is the same.
Area 1 1024 points Area 2 512 points Area 3 256 points Area 4 128 points
a Find the probability that a dart randomly hitting the board will score:
i 64
ii a multiple of 128
iii a number from 16 to 256 inclusive
iv a number from 17 to 1023 inclusive or a number less than 256.
b Why is it necessary to state that the areas bound by any two consecutive circles are the same?
11E
karnaugh maps and probability
tables
We have seen how Venn diagrams provide a visual representation of sets and probabilities. Another effective approach is to display the information by means of a Karnaugh map. Consider a Venn diagram for two sets A and B.
ξ
A B
A∩ B' A ∩ B A' ∩ B A' ∩ B'
Notice that the Venn diagram consists of four mutually exclusive regions, A ∩ B′, A ∩ B, A′ ∩ B and
A′ ∩ B′. These four subsets of ξ can be presented as a Karnaugh map.
Column 1 Column 2 Column 3
B B′ Row 1 A A ∩ B A ∩ B′ Row 2 A′ A′ ∩ B A′ ∩ B′ Row 3 diGiTal doC doc-9807 WorkSHEET 11.2
Comparing the table entries with the Venn diagram provides equality relationships across rows and down columns. That is, in terms of regions we can see that for column 1, (A ∩ B) ∪ (A′ ∩ B) = B, and for column 2, (A ∩ B′) ∪ (A′ ∩ B′) = B′.
Similarly, for row 1, (A ∩ B) ∪ (A ∩ B′) = A, and for row 2, (A′ ∩ B) ∪ (A′ ∩ B′) = A′. The third row and column can be used to check the sum totals of each row and column. This type of verifi cation can be useful in practical problems.
The probability table
We can present a Karnaugh map in terms of the probability of each of the four subsets A ∩ B′, A ∩ B,
A′ ∩ B and A′ ∩ B′ of ξ.
B B′
A Pr(A ∩ B) Pr(A ∩ B′) Pr(A)
A′ Pr(A′ ∩ B) Pr(A′ ∩ B′) Pr(A′) Pr(B) Pr(B′) 1
Note the value of 1 at the bottom right of the table. This is the sum of the probabilities across the last row and the sum of the probabilities down the last column.
That is, Pr(B) + Pr(B′) = 1 and Pr(A) + Pr(A′) = 1.
Consider the following example. A survey of 1000 taxi drivers revealed that 450 of them drive Falcons and 500 drive Commodores. It was also found that 350 taxi drivers have occasion to use both types of car. This information can be represented as a Venn diagram, a Karnaugh map or a probability table. ξ F C 150 350 100 400 C C′ Row 1 F 350 100 450 Row 2 F′ 150 400 550 Row 3 500 500 1000 C C′ F 0.35 0.1 0.45 F′ 0.15 0.4 0.55 0.5 0.5 1.0
Venn diagram Karnaugh map Probability table
The Karnaugh map provides the following information:
1. 350 drivers drive both a Falcon and a Commodore (row 1, column 1: F ∩ C ). 2. 100 drivers drive only a Falcon (row 1, column 2: F ∩ C ′).
3. The total number of Falcon drivers is 450 (350 + 100).
4. 150 drivers drive only a Commodore (row 2, column 1: F ′ ∩ C ).
5. 400 drivers do not drive either a Falcon or a Commodore (row 2, column 2: F ′ ∩ C ′). 6. There are 500 Commodore drivers altogether (350 + 150).
7. There are 1000 drivers in total (row 3, column 3).
WorkEd ExamplE 18
Complete the probability table shown below and represent the information as a Venn diagram.
Column 1 Column 2 Column 3
B B′ Row 1 A 0.3 Row 2 A′ 0.25 Row 3 0.65 1 TUTorial eles-1450 Worked example 18
Think WriTE/draW
1 Find the value for row 2, column 3 and
for row 3, column 2. B B′
A 0.3
A′ 0.25 0.7 0.65 0.35 1
2 Find the value for row 2, column 1 and for row 1,
column 2. A B 0.1B′ 0.3
A′ 0.45 0.25 0.7 0.65 0.35 1
3 Find the value for row 1, column 1. B B′
A 0.2 0.1 0.3
A′ 0.45 0.25 0.7 0.65 0.35 1
4 Represent the information as a Venn diagram. ξ
A B 0.45 0.2 0.1 0.25 WorkEd ExamplE 19
Complete a probability table, given that Pr(A′ ∩ B) = 0.24, Pr(A) = 0.32 and Pr(B) = 0.35.
Think WriTE
1 Place the known information in the
appropriate cells of the probability table. B B′
A 0.32
A′ 0.24
0.35 1
2 Build up the table using the given information and the fact that the probability totals 1.
B B′ A 0.11 0.32 A′ 0.24 0.68 0.35 0.65 1 B B′ A 0.11 0.21 0.32 A′ 0.24 0.44 0.68 0.35 0.65 1 WorkEd ExamplE 20
A group was surveyed in relation to their drinking of tea and coffee. From the results it was established that if a member of the group is randomly chosen, the probability that that member drinks tea is 0.5, the probability that they drink coffee is 0.6, and the probability that they drink neither tea nor coffee is 0.1.
a Use the information to complete a probability table.
b Calculate the probability that a randomly selected person of the group: i drinks tea but not coffee
Think WriTE a 1 Let T and C be the set of people who drink
tea and coffee respectively. Place the given information in the table.
a C C′ T 0.5 T′ 0.1 0.6 1 C C′ T 0.2 0.3 0.5 T′ 0.4 0.1 0.5 0.6 0.4 1
2 Build up the table as shown.
C C′ T 0.5 T′ 0.1 0.5 0.6 0.4 1 C C′ T 0.3 0.5 T′ 0.4 0.1 0.5 0.6 0.4 1
b 3 Use the appropriate probability from the
table. b iii Pr(T ∩ C′ ) = 0.3Pr(T ∩ C ) = 0.2
Exercise 11E
karnaugh maps and probability tables
1 WE 18 Complete each Karnaugh map and represent the information as a Venn diagram.
a B B′ A 17 25 A′ 13 15 b B B′ A 33 A′ 27 72 114 c B B′ A 0.3 0.57 A′ 0.4 d B B′ A 0.03 A′ 0.22 0.36
2 mC Decide which of the following statements is true.
a U= 0.15 B V + W = 0.42 C X + Y = 0.55
d V − X = W − 0.58 E U + Z = W − Z
3 Complete a Karnaugh map given that n(A ∩ B) = 87, n(A′ ∩ B) = 13, n(A ∩ B′) = 63 and
n(ξ ) = 218.
4 Complete a Karnaugh map given that n(A ∩ B) = 35, n(A ∩ B′) = 29, n(A′ ∩ B′) = 44 and
n(A′ ∩ B) = 56.
5 Draw a Karnaugh map representing each Venn diagram.
a ξ A B 0.3 0.11 0.45 0.14 b ξ A B 0.12 0.61 0.27 B B′ A 0.31 Y 0.75 A′ X U Z V 0.58 W
c ξ A B 4 6 10 d A B 18 5 7 15 ξ A B 18 5 7 15
6 Determine the probability values and complete a probability table using the given information.
a ξ = {letters of the alphabet from a to k}, A = {a, b, c, d, e, f, g}, B = {e, f, g, h}
b ξ = {fi rst 20 natural numbers}, A = {natural numbers less than 11}, B = {natural numbers from 8 to 15 inclusive}
7 mC If A = {2, 7, 8, 10}, B = {3, 5, 7, 9, 10} and ξ = {1, 2, . . . , 10}, then A ∩ B′ will contain the set: a {3, 5, 7} B {1, 4} C {2, 6, 9}
d {6, 7, 10} E {2, 8}
8 A survey of students revealed that 30 of them like football, 26 like soccer, 6 like both sports and 10 prefer a sport other than football or soccer. Represent this information as a:
a Venn diagram b Karnaugh map.
9 Complete a probability table for the information in question 8.
10 mC Of a group of 200 people, 48% drink coffee (C ) each day and 39% drink tea (T ). If 38% of the
people do not drink tea or coffee, the probability table is:
a T T′ C 0.25 0.23 C′ 0.14 0.38 B T T′ C 0.23 0.14 C′ 0.38 0.25 C T T′ C 0.25 0.35 C′ 0.48 0.38 d T T′ C 0.39 0.38 C′ 0.48 0.23 E T T′ C 0.38 0.10 C′ 0.01 0.51
11 WE 19 Complete a probability table, given:
a Pr(A ∩ B) = 0.3, Pr(A′ ∩ B′) = 0.2 and Pr(A) = 0.6 b Pr(B ∩ A′) = 0.7, Pr(B) = 0.8 and Pr(B′ ∩ A) = 0.1 c Pr(A ∩ B) = 0.5, Pr(A′ ∩ B′) = 0.1 and Pr(B′) = 0.4
d Pr(A′ ∩ B) = 1 4, Pr(A ∩ B′) = 1 4 and Pr(B′) = 3 4.
12 Two hundred and eighty children were asked to indicate their preference for ice-cream flavours. It was found that 160 of the children like chocolate flavour, 145 like strawberry and 50 like both flavours. Use this information to complete a Karnaugh map.
13 WE20 An examination of 250 people showed that of those in the group who are less than or equal to 20 years of age, 80 wear glasses and 55 do not. Also, 110 people over 20 years of age must wear glasses.
a Represent the information as a probability table.
b Calculate the probability that a randomly selected person of the group:
i does not wear glasses and is over 20 years of age
ii is 20 years of age or younger.
14 For the probability table shown, A is the event ‘no more than 15 years of age’ and B is the event ‘smoker’.
a Complete the probability table.
b What is the probability that:
i a person older than 15 years of age does not smoke?
ii a person is a smoker and is older than 15 years of age?
iii the person is a smoker over the age of 15 or is a non-smoker less than or equal to 15 years of age?
B B′
A 0.08
A′ 0.6
15 A survey of a group of business people indicates that 42% of those surveyed read the Daily Times newspaper only each day and 18% read both the Daily Times and the
Bugle. Additionally, 12% of those questioned stated that they do not read either of these newspapers.
a Show this information as a Karnaugh map.
b What is the probability that a randomly selected member of the group:
i does not read the Daily Times?
ii reads the Bugle only?
iii does not read the Bugle or does not read either newspaper?
c If the group consists of 150 business people, determine how many members read at least one newspaper.
16 A lucky dip box contains 80 marble tokens that can be exchanged for prizes. Some of the marbles have a red stripe, some have a blue stripe, some have both a red and a blue stripe, and some marbles have no stripes at all. It is known that 25% of the marbles have a red stripe on them, 20% of them have a blue stripe and 65% have no stripe.
a Present the information as a Karnaugh map.
b What is the probability of choosing a marble that has a red stripe only?
c Find the probability of choosing a marble that has a red and a blue stripe or no stripe.
11F
Conditional probability
Erin thinks of a number from 1 to 10 (say 8) and asks Peter to guess what it is. The probability that Peter makes a correct guess on his first try is 1
10. If, however, Erin first tells Peter that the number is greater
than 7, his chances are better because he now knows that the number must be one of the numbers 8, 9 or 10. His probability of success is now 1
3.
This problem may be stated as: What is the probability of Peter choosing the right number from 1 to 10, given that the number is greater than 7?
This is an example of conditional probability, where the probability of an event is conditional on (that is, it depends on) another event occurring first. The effect in this case is to reduce the event space and thus increase the probability of the desired outcome.
For two events A and B, the conditional probability of event A given that event B occurs is denoted by Pr(A | B) and is given by:
Pr(A | B) = Pr(A ∩ B)
Pr(B) , Pr (B) ≠ 0 Event B is sometimes called the reduced event space.
For the example above, if we let B be the event ‘numbers greater than 7’ and A be the event ‘Erin’s secret number’, then we may write:
Pr(A | B) = Pr(A ∩ B) Pr(B) = 1 10 3 10 = (1 10÷ 3 10) = 1 3
The reduced event space can be illustrated by the Venn diagram below.
ξ A B 8 9 10 1 2 3 4 5 6 7
WorkEd ExamplE 21
If Pr(A ∩ B) = 0.8 and Pr(B) = 0.9, find Pr(A | B).
Think WriTE
Substitute the values given into the expression for
conditional probability. Pr(A | B) =
Pr(A ∩ B) Pr(B) = 0.8 0.9 = 8 9 WorkEd ExamplE 22
If Pr(A) = 0.3, Pr(B) = 0.5 and Pr(A ∪ B) = 0.6, calculate:
a Pr(A ∩ B) b Pr(A | B).
Think WriTE
a Use the Addition Law for probabilities
to find Pr(A ∩ B). a Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B)0.6 = 0.3 + 0.5 − Pr(A ∩ B) so Pr(A ∩ B) = 0.2
b Use the formula for conditional probability to
find Pr(A | B). bPr(A | B) =
Pr(A ∩ B) Pr(B) = 0.2 0.5 = 2 5 WorkEd ExamplE 23
Of a group of 50 Year 11 students, 32 study Art and 30 study Graphics. Each student studies at least one of these subjects.
a How many students study both?
b Illustrate the information as a Venn diagram.
c What is the probability that a randomly selected student studies Art only?
d Find the probability that a student selected at random from the group studies Graphics, given that the student studies Art.
Think WriTE/draW
a 1 Define relevant events. a Let A = students who study Art
G = students who study Graphics
A ∩ G = students who study both
2 Find the number who study both subjects
using set theory. n(A ∪ G) = n(A) + n(G) − n(A ∩ G)50 = 32 + 30 − n(A ∩ G) 50 = 62 − n(A ∩ G)
n(A ∩ G) = 12
b Show all the information on a Venn diagram. bξ
Art Graphics 20 12 18
c The Venn diagram reveals that 20 of the 50 students
study Art only. Calculate the probability. c Pr(Art only) =
20 50
= 2 5
d Use the conditional probability formula to fi nd the probability that a student studies Graphics, given that the student studies Art.
dPr(G | A) = Pr(G ∩ A) Pr(A) = 12 50 ÷ 32 50 = 12 32 = 3 8 WorkEd ExamplE 24
Seated in a Ford Falcon are 4 males and 2 females. Seated in a Holden Commodore are 2 males and 1 female. One of the cars is randomly stopped by the police and one person from the vehicle is randomly selected.
Draw a tree diagram to illustrate the situation and calculate the probability that:
a the person selected by the police is female
b if a female is selected by the police, she was sitting in the Ford.
Think WriTE/draW a 1 Calculate the probabilities. aPr(Ford) = 1 2 Pr(Holden) = 1 2
Pr(male from Ford) = 4
6 Pr(male from Holden) = 2 3
= 2
3
Pr(female from Ford) = 2
6 Pr(female from Holden) = 1 3
= 1
3 2 Draw the tree diagram.
male Ford and male female Ford and female
male Holden and male female Holden and female Holden Ford Car Person 1– 2 1– 2 2–3 1– 3 2– 3 1– 3
3 Use the tree diagram to work out the probability that the person is female. Consider all the ways a female may be selected.
Pr(female selected) = Pr(Ford and female or Holden and female) = Pr(Ford and female) + Pr(Holden and female) = 1 2× 1 3+ 1 2× 1 3 = 1 3
b Use the tree diagram and the formula for conditional probability.
bPr(person is from the Ford | female is selected)
= = × =
Pr(Ford and female) Pr(female) 1 2 13 1 3 1 2 TUTorial eles-1451 Worked example 24
Once the tree diagram was drawn, the calculation for part a in worked example 24 was quite intuitive. In order to calculate the probability of a female being selected, the occupants of both cars needed to be considered. In fact, there is a rule of probability that formalises the calculation performed in part a above. The law is known as the Law of Total Probability, and it states:
Pr(A) = Pr(A | B)Pr(B) + Pr(A | B′)Pr(B′)
To calculate the answer to part a of worked example 24, let Pr(A) = Pr(female) and Pr(B) = Pr(Ford).
Pr(female) = Pr(female given the car is a Ford)Pr(Ford) + Pr(female given the car is a Holden)Pr(Holden) = 1 3× 1 2+ 1 3× 1 2 = 1 3
Note that the Law of Total Probability simplifi es to give the rule used in part a of worked example 24: Pr(A) = Pr(A | B)Pr(B) + Pr(A | B′)Pr(B′)
Pr(A) = Pr(A ∩ B) + Pr(A ∩ B′)
Exercise 11F
Conditional probability
1 WE21 If Pr(A) = 0.8, Pr(B) = 0.5 and Pr(A ∩ B) = 0.4, find: a Pr(A | B) b Pr(B | A).
2 If Pr(A) = 0.65, Pr(B) = 0.75 and Pr(A ∩ B) = 0.45, find:
a Pr(A | B) b Pr(B | A).
3 If Pr(A ∩ B) = 0.4 and Pr(A) = 0.5, find Pr(B | A). 4 If Pr(A ∩ B) = 0.25 and Pr(B) = 0.6, find Pr(A | B).
5 If Pr(B | A) = 0.32 and Pr(A) = 0.45, find Pr(A ∩ B). 6 If Pr(A | B) = 0.21 and Pr(B) = 0.8, what is Pr(A ∩ B)?
7 Calculate Pr(A) if Pr(B | A) = 0.75 and Pr(A ∩ B) = 0.5.
8 Calculate Pr(B) if Pr(A | B) = 0.96 and Pr(A ∩ B) = 0.8.
9 WE22 If Pr(A) = 0.7, Pr(B) = 0.5 and Pr(A ∪ B) = 0.9, calculate: a Pr(A ∩ B)
b Pr(B | A).
10 mC If Pr(B | A) = 0.8 and Pr(A ∩ B) = 0.6, then Pr(A) is: a 4 5 B 3 5 C 1 4 d 3 4 E 2 3
11 mC If Pr(A) = 0.9 and 2 × Pr(A ∩ B) = Pr(A), then Pr(B | A) is:
a 12 B 59 C 25
d 19 E 49
12 Show that if Pr(A ∩ B) = Pr(A) × Pr(B), then Pr(B | A) = Pr(B).
13 If Pr(A) = 0.23, Pr(B) = 0.27 and Pr(A ∪ B) = 0.3, find:
a Pr(A ∩ B) b Pr(A | B).
14 If Pr(A) = 0.45, Pr(B) = 0.52 and Pr(A ∪ B) = 0.67:
a fi nd Pr(A ∩ B)
b fi nd Pr(B | A)
c represent the information as a Venn diagram.
15 A box contains marbles numbered 1, 2, 3, . . . 50. One marble is randomly taken out of the box. What is the probability that it is:
a a multiple of 3, given that it is less than 21?
b between 11 to 39 inclusive, given that it is greater than 20?
diGiTal doC doc-9808 SkillSHEET 11.1 Conditional probability
16 mC A group of 80 females consists of 54 dancers and 35 singers. Each member of the group is either a dancer or a singer or both. The probability that a randomly selected member of the group is a singer given that she is a dancer is:
a 0.17 B 0.44
C 0.68 d 0.11
E 0.78
17 WE23 A group of 60 adventurers comprises 30 mountain climbers and 45 scuba divers. If each adventurer does at least one of these activities:
a How many adventurers are both climbers and divers?
b Illustrate the information on a Venn diagram.
c What is the probability that a randomly selected group member is a scuba diver only?
d Find the probability that an adventurer randomly selected from the group is a scuba diver, given that the adventurer is a mountain climber.
18 Of 200 families surveyed, 85% have a TV and 70% possess a CD player. Assuming each family has at least one of these items, what is the probability that one family randomly selected has a TV, given that they also own a CD player?
19 During the Christmas holidays 42 students from a group of 85 VCE students found vacation employment while 73 students went away on holidays. Assuming that every student had at least a job or went on a holiday, what is the probability that a randomly selected student worked throughout the holidays (that is, did not go away on holidays), given that he/she had a job?
20 WE24 The probability that a machine in a chocolate factory does not coat a SNAP chocolate bar adequately, therefore producing a defective product, is 0.08. The probability that it does not coat a BUZZ chocolate bar adequately is 0.11. On any day the machine coats 250 SNAP bars and 500 BUZZ bars. A chocolate bar is chosen at random from the production line. Draw a tree diagram to illustrate the situation and find the probability that the chocolate bar chosen at random is:
a a BUZZ chocolate bar
b a SNAP chocolate bar and is defective
c defective, given that a SNAP bar is chosen.
21 The staff at Happy Secondary College is made up of 43 females and 29 males. Also, 22% of the females are under 40 years old, and 19% of the males are under 40. If a staff member is selected at random, what is the probability that:
a a male is selected?
b a male 40 years or over is selected?
c a female under the age of 40 is selected?
d a person under 40 years of age is selected?
e the person is a female given that the person selected is under 40 years of age?
22 Two letters are randomly picked from the word INFINITESIMAL. If a letter can be used more than once, calculate the probability that both letters selected are vowels, given that the first letter is a vowel.
11G
Transition matrices and
markov chains
introduction
In chapter 7 we saw many uses for matrices, from displaying information in an organised manner to solving simultaneous equations or representing transformations. Matrices are also very useful in certain conditional probability problems.
inTEraCTiViTY int-0270
