21
24--- 34
51--- 39
91--- 13
16--- 27
30
---I tossed a coin 4 times and got 1 head.
That means the probability of getting
a head must be 1 in 4!
. . . But I got 3 heads when I tried that!
Performing an experiment will not always give a consistent result, or even a result we may think is most likely to occur.
In many cases we can work out the expected or theoretical probability of an event by considering the possible outcomes. For example, when tossing a coin there are two possible outcomes, a head or a tail.
Since there is only one head, the probability of throwing a head would be 1 out of 2, ie .
Worked examples
1 If a dice is rolled, what is the probability of getting:
a a six? b an odd number? c a number less than seven?
2 In a bag there are six blue marbles, four white marbles and two red marbles. What is the probability of choosing at random:
a a blue marble? b a blue or white marble? c a pink marble?
Solutions
1 The possible outcomes when rolling a dice are 1, 2, 3, 4, 5, 6. So the number of possible outcomes is 6.
a The number of sixes on a dice is 1. b The number of odd numbers on a dice So the probability of throwing a is 3. So the probability of throwing an six is 1 out of 6, or . This can be odd number is 3 out of 6.
written as: P(odd no.) =
P(6) = =
c Since all six numbers on a dice are less than seven, the probability of throwing a number less than seven is:
P(no. < 7) =
= 1
2 The total number of marbles in the bag is twelve. So the number of possible outcomes is 12.
a Number of blue marbles is six. b Number of blue or white marbles is ten.
∴ P(blue marble) = ∴ P(blue or white) =
= =
c Number of pink marbles is zero.
∴ P(pink) =
= 0
1 2
---Drawing a picture often helps.
■ When calculating the probability of an event we shall assume that each possible outcome is equally likely, ie no two-headed coins or loaded dice.
1 6
---3 6 ---1
6--- 1
2
---■ The probability of an event certain to happen is 1.
P(sure thing) = 1
6 6
---6
12--- 10
12 ---1
2--- 5
6
---■ The probability of an event that cannot happen is 0.
P(impossibility) = 0
0 12
---It must be pointed out that the probabilities of each possible event must add up to 1. As a consequence of this, if the probability of an event occurring is P(E), then the probability of E not occurring is 1 − P(E).
E′ is set notation for the ‘complement’ of E, ie those outcomes outside of E. For example:
• The complementary event for getting an even number after rolling a dice is getting an odd number.
• The complementary event for drawing a red card from a deck of cards is drawing a black card.
A single dice is thrown. What is the probability of getting:
a a one?
b an even number?
c a number less than 3?
Ten coloured discs are placed in a hat. Five are red, three are yellow and two are black. If one disc is drawn from the hat, what is the probability that the disc will be:
a red? b black? c red or black?
d not black? e blue? f red, yellow or black?
For each event given here, write the complementary event.
a Getting an odd number after a dice is thrown.
b Getting a tail when a coin is tossed.
c Getting a number less than 6 when a dice is thrown.
d Drawing a spade from a standard deck of playing cards.
e Seeing red displayed on a traffic light that is working.
f Winning a soccer match.
g Choosing a vowel from the letters of the alphabet.
If each possible outcome is equally likely, then:
probability of an event, P(E) = where n(E) = number of ways the event can occur
n(S) = number of ways all events can occur
(S is used to represent the sample space, which is the set of possible outcomes.) The probability of any event occurring must lie in the range 0 P(E) 1.
n E( ) n S( )
---■ P(E′) = 1 − P(E) where P(E′) is the probability of E not occurring.
Exercise 4:03
Theoretical probability NS5·1·3 1 A coin is tossed.Find the probability of:
a head b tail
2 Four cards are labelled A, B, C, D.
Find the probability of selecting the:
a A b B or C
Foundation Worksheet 4:03
1
2
3
From a standard pack of 52 playing cards, a card is drawn at random. What is the probability that the card is:
a the Ace of diamonds? b a King?
c a red card? d a spade?
e a black Jack? f a 7 or 8?
g a picture card (Jack, Queen or King)?
The 26 letters of the alphabet are written on cards and placed in a box. If one card is picked at random from the box, what is the chance that the letter on it will be:
a X? b a vowel?
c M or N? d a letter in the word MATHEMATICS?
Stickers were placed on a dice so that the faces showed three 2s, two 4s and a 6. If the dice is now thrown, what is the probability that the upper surface will be:
a a 2? b a 4? c a 6?
d even? e odd? f less than 6?
If the probability of an event is , how many times, on average, would you expect it to occur in 20 trials? Can you say for certain how many times it will occur?
A bag contains five red balls with the numbers 1 to 5 painted on them and seven blue balls painted with the numbers 1 to 7. If a ball is chosen at random, what is the chance of choosing:
a a red ball? b a ball numbered 3?
c a ball numbered 6? d the blue ball numbered 1?
e an even numbered ball?
If a dice is rolled twelve times, how many times, on average, would you expect the result to be:
a a six? b an odd number? c a number greater than 2?
A survey of Kylie Crescent shows that 20% of the families have 1 child, 45% have 2 children, 15% have 3 children and 10% have more than 3 children. What is the probability that a family chosen at random will have:
a 2 children? b more than 2 children?
c at least 2 children? d no children?
A roulette wheel is numbered from 0 to 36.
Half of the numbers from 1 to 36 are red, the other half are black and the zero slot is green. Find the probability that the result of a spin will be:
a odd b black
c 0 d not red
e a number from 1 to 6 inclusive f a number greater than 25
If the result is the green zero, the bank wins all wagers made.
g How often would you expect a zero?
The payout for black or red is ‘even money’
(ie for a successful bet of $1, the payout is $1 plus the original $1 bet).
h If a gambler continued to bet the same amount on black, should he or she eventually win, 4
5
6
7 1
5
---8
9
10
My interest in this is purely mathematical.
11
A bag contains 5 red, 10 blue and 15 green marbles. I am asked to choose a ball from the bag. Before choosing the ball I am allowed to add or subtract 5 balls of the same colour.
What would you do to give yourself the best chance of selecting:
a a blue ball? b a green ball?
Steven is told that cup A contains 3 red and 5 white marbles, while cup B contains 6 red and 4 white marbles.
He has to select a ball from either cup, but before he does he has to say what colour it will be. He wins if he selects a ball of the colour that he has chosen. He is told that before selecting a ball, he can, if he wishes, select a ball of any colour from either cup and place it in the other
cup. What should Steven do to give himself the greatest chance of winning?
Kate has 16 cards: 8 are red and 8 are black. She has been asked to divide them into two piles so that the probability of selecting a red card from one pile is twice the probability of selecting a red card from the other pile. She does not have to use all the cards. How could she divide the cards? Is there more than one solution?
A bag contains 7 red and 18 white balls. How many red balls must be added to the bag so that the probability of choosing a red ball from the bag is:
a ? b ? c ?
12
cup A cup B
13
14
15
1
2--- 3
4--- 9
10
---• One of these keys is the one I want. If I choose one of them at random, what is the probability that it is the one I want?
• If the first one I chose was not the one I wanted, what would be the probability that I will choose the right one on my second choice?
• Use experimental probability to find the chance of choosing the correct key on either my first or second selection.
You can use a spreadsheet program such as Excel to simulate the throwing of a dice.
• Open the spreadsheet program and using the mouse, click on the first cell A1.
• Type in the formula: = INT(RAND()*6+1)
• Press ENTER and a number from 1 to 6 will appear in cell A1.
• Using the mouse, click on the bottom right hand corner of cell A1 and drag the cursor down to, say, cell A12. Release the button on the mouse and random numbers from 1 to 6 will appear in these cells.
• If these cells are again highlighted and the corner of cell A12 is dragged to, say, C12, more random numbers will appear in the cells in columns B and C.
1 Examine these results to see how closely they represent the expected probabilities when a dice is thrown.
2 Use the spreadsheet with 3 columns of random numbers to work out your own answer to question 11 in Exercise 4:02.
(Note: the function RAND() tells the computer to generate a random number between 0 and 1. Multiplying by 6 and adding 1 converts this to a number greater than 1 and less than 7. The function INT tells the computer to ‘cut off’ the decimal part of the number
Challenge 4:03 Computer dice
1