Pearson Mathematics 8 SB COVER - Weebly

556 PEARSON mathematics 8

Understanding probability

An event in probability is simply something that occurs. Examples of events in probability are tossing a head with a coin, rolling a 4 on a die or having a day when it rains.

The probability of an event occurring is the chance of that event occurring. Probabilities range from 0, where the chance of an event is impossible, to 1, where the chance of the event occurring is certain. The probability of obtaining a 7 on a normal die would be 0, whereas the probability of getting a number less than 7 would be 1.

Any possible result of an event is called an outcome. When tossing a coin, there are two possible outcomes: obtaining a head or obtaining a tail. A successful outcome is the result that is required, such as getting a head on a coin toss. The probability of an event, Pr(event), can be written as a decimal, fraction or percentage.

Probabilities can be determined through experiments and by collecting long-term data, but we often determine probabilities from theory.

Sample spaceTo calculate a theoretical probability, we need to know the sample space. A sample space is a list of all possible outcomes and the outcomes must all be equally likely.

Successful outcomes must be some or all of the outcomes listed in the sample space. The sample space from tossing a coin is a head (H) and a tail (T). If a successful outcome is tossing a head, there is only one successful outcome possible out of two outcomes in the sample space, so:

Pr(H) =

The sample space from rolling a normal die is 1, 2, 3, 4, 5, 6. If a successful outcome is rolling a 2, then there is only one successful outcome possible out of six outcomes in the sample space, so:

Pr(2) =

When more than one outcome is considered a success, we can calculate the probability by counting the number of successes overall or calculating each probability separately and adding them.

For example, finding the probability of rolling a 1 or a 2 on a die can be calculated in two ways.

There are two successful outcomes possible out of six outcomes in the sample space, so:

Pr(1 or 2) = =

However, we could have calculated this by considering the probability of each event separately and adding them:

Pr(1) = and Pr(2) = so Pr(1 or 2) = +

=

=

12---

16---

26--- 1

3---

16--- 1

6--- 1

6--- 1

6---

26---

13---

9.5AA Pearson Maths 8 SB-09.fm Page 556 Wednesday, August 10, 2011 2:26 PM

9 Statistics and probability

9.5

557

Calculating probability

ComplementsAll situations can be written so that there are only two possible outcomes. An event may happen or it may not happen. These two probabilities add up to 1. The outcome of a game of football could be win, lose or draw, but we could consider winning and not winning as the two outcomes. One outcome is the complement of the other. The probabilities of ‘winning’ or ‘not winning’ are called complementary probabilities.

Pr(event) =

All outcomes must be equally likely.

Worked Example 12In Kayla’s class there are 24 students. Five were born in Cambodia, three in Vietnam, six in Australia, two in Croatia, five in Serbia, one in England, one in New Zealand and one in South Africa. A member of the class is selected at random. What is the probability that a student chosen at random was born in:

(a) Vietnam (b) Australia (c) Croatia or Serbia?

Thinking(a) 1 Write the number of successful

outcomes.(a) Number born in Vietnam = 3

2 Write the total number of outcomes. Total = 24

3 Use the formula to express the probability as a fraction and then cancel down to simplest terms (if possible).

Pr(born in Vietnam) =

=

(b) 1 Write the number of successful outcomes.

(b) Number born in Australia = 6



Pr(born in Australia) =

=

(c) 1 Calculate the number of successful outcomes.

(c) Number born in Croatia or Serbia = 2 + 5= 7



Pr(born in Croatia or Serbia) =

Note: if we added together each of the individual probabilities for country of birth they would have to equal 1. It is certain that each student was born in one of the listed countries.

number of successful outcomestotal number of outcomes

-------------------------------------------------------------------------------

12

324------

18---

624------

14---

724------

AA Pearson Maths 8 SB-09.fm Page 557 Wednesday, August 10, 2011 2:26 PM

9.5


Sometimes, the probability you want to calculate is easier to find if you calculate the probability of the complement and subtract this from 1.

Pr(event not occurring) = 1 − Pr(event occurring)

Consider the probability of getting a 2, 3, 4, 5 or 6 when rolling a die. This probability is the same as the probability of ‘not getting a 1’, so if we calculate the probability of ‘getting a 1’ and subtract this from 1 we will get the desired probability.

Pr(event occurring) + Pr(event not occurring) = 1

Worked Example 13If I roll an unbiased six-sided die, what is the probability of:

(a) not rolling a 6 (b) not rolling a 5 or a 6?

Thinking

(a) 1 Write the known probability. (In this case, the probability of rolling a 6.)

(a) Pr(rolling a 6) =

2 Use the formula Pr(event not occurring) = 1 − Pr(event occurring) and calculate the answer.

Pr(not rolling a 6) = 1 − Pr(rolling a 6)

= 1 −

=

(b) 1 Write the known probabilities. (b) Pr(rolling a 5) = and

Pr(rolling a 6) =

2 Find the total known probability. Pr(rolling a 5 or a 6) = +

=

=

3 Use the formula Pr(event not occurring) = 1 − Pr(event occurring) and calculate the answer.

Pr(not rolling a 5 or a 6) = 1 −

=

13

16---

16---

56---

16---

16---

16--- 1

6---

26---

13---

13---

23---



9.5

559

Understanding probability

Fluency1 A class consists of 26 students. Ten study Chinese and the rest study Italian. What is the

probability that a student chosen at random studies:

(a) Chinese (b) Italian?

2 If I spin a five-sided spinner with numbers 1, 2, 3, 4, 5, what is the probability of:

(a) not getting a 2

(b) not getting a 1 or a 2?

3 Illona is running in a race at the local athletics track. If her probability of winning is what is the probability that she will not win?

4 There are 13 teachers in the staff room: eight females and five males, one of whom is Mr Antonac. A student knocks at the door wanting to see Mr Antonac. What is the probability that the teacher who answers the door is:

(a) male (b) female (c) Mr Antonac?

5 The hockey training squad consists of eight forwards, five half-backs, four full-backs and Jarrod, who plays in goal. One player is chosen at random from the training squad to represent the team at a fundraising function.

What is the probability that the person is:

(a) a forward (b) a half-back (c) Jarrod, the goalkeeper?

6 There are 10 000 tickets sold in the school raffle. If Alf bought 10 tickets, find:

(a) Pr(Alf wins the raffle) (b) Pr(Alf does not win the raffle).

7 My fruit bowl contains five bananas, four apples, one orange and five peaches. If I choose one piece of fruit from the bowl at random, find:

(a) Pr(a banana)

(b) Pr(a peach or an apple)

(c) Pr(not an apple)

(d) Pr(not an orange or a banana).

NavigatorQ1, Q2, Q3, Q4, Q5, Q6, Q7, Q8, Q10, Q11, Q12, Q15, Q16, Q17

Q1, Q2, Q3, Q4, Q5, Q6, Q7, Q8, Q10, Q11, Q12, Q13, Q14, Q15, Q17

Q1, Q2, Q3, Q4, Q5, Q6, Q7, Q8, Q9, Q11, Q12, Q13, Q14, Q15, Q16, Q17

9.5

Answerspage 650

12

1334

5 2

1

18---


9.5


8 A bag of potatoes contains five green ones and fourteen good ones. If a potato is chosen at random, the probability of it being green is:

A B C D

Understanding9 What is the probability that the first digit on a car number plate will be a 2, 3 or 4? (Ignore

the possibility of personalised number plates without digits.)

10 For seven of the last twelve years, Dennis has received a birthday present of socks and handkerchiefs from his grandparents. Assuming his grandparents don’t change their giving behaviour, what is the probability that he doesn’t receive socks and handkerchiefs from his grandparents for his next birthday?

11 Dominic rolls a standard six-sided die. Find:

(a) Pr(a 2 or a 4) (b) Pr(a number larger than 2)

(c) Pr(a number less than 2) (d) Pr(a multiple of 3)

(e) Pr(not a 5) (f) Pr(not a number larger than 3).

12 A bag contains 26 small tiles, each marked with a different letter of the alphabet. Celeste selects one tile from the bag. Choose the answer with the correct probability that the tile Celeste draws is:

(a) a vowel

A B C D

(b) b, d, g or z

A B C D

(c) one of the letters in the word ‘chance’.

A B C D

13 The numbers from one to twenty are written on separate counters and placed in a hat. If one counter is drawn at random, find:

(a) Pr(a number between 5 and 11) (b) Pr(a number with a 1 in it)

(c) Pr(a number less than 30) (d) Pr(a multiple of 5)

(e) Pr(not a multiple of 5) (f) Pr(not odd).

Reasoning14 The following table shows the

number of vehicles sold across Australia in March 2010 divided into a number of vehicle types.

519------ 5

14------ 1

2--- 14

19------

213------ 5

26------ 1

5--- 1

4---

213------ 4

25------ 3

13------ 1

4---

213------ 5

26------ 3

13------ 6

25------

Source: VFacts (official results issued by the Federal Chamber of Automotive Industries)

Vehicle type Number sold in March 2010

Light carSmall carMedium carLarge carUpper large carPeople moverSports carSUV CompactSUV MediumSUV LargeSUV Luxury

12 15322 463

72568816272

123415239651869014071895



9.5

561

Express your answers to the following questions as percentages, correct to one decimal place. Find the probability that a new vehicle buyer in March 2010, chosen at random from purchasers of vehicles in the table, had purchased a:

(a) light car

(b) small car

(c) medium car

(d) SUV of any type

(e) not a people mover

(f) not a sports car nor a people mover.

(g) Do you think that the probabilities would change very much from year to year? Why or why not?

(h) Is it more probable that a new vehicle buyer in March 2010 bought a small car or an SUV of any type? Justify your answer.

15 (a) Draw a spinner for which Pr(green) = Pr(blue) = Pr(orange) =

(b) Draw a spinner for which Pr(red) = 3 × Pr(blue) and Pr(green) =

(c) Draw a spinner for which Pr(brown) = 5 × Pr(green), Pr(red) = 3 × Pr(green) and Pr(yellow) = Pr(green).

Open-ended16 (a) Look at the following list of words and expressions very carefully and then put them

in order from impossible on the left to certain on the right. Then, assign a percentage probability to each of them. Your teacher may ask you to discuss your order with some other members of the class.

50-50 certain even chanceimpossible little chance more often than notmost likely most often most unlikelyprobably sure thing unlikelyunusual usually very unlikely

(b) Can you add some other words or expressions to the list?

17 Assign a probability to the following situations in your life.

(a) getting to school on time every day (b) having an apple for lunch

(c) going to the movies at least once a month (d) getting a phone call from a friend today

(e) running a marathon in your lifetime (f) taking a space ride to the Moon

(g) playing sport on the weekend (h) sleeping after 9 a.m. on a Saturday

13--- .

15--- .

PuzzleBracket placementInsert brackets to make each of the following equations true.

1 3 × 4 − 5 + 6 × 2 + 5 = 39

2 5 − 2 × 3 − 4 ÷ 6 − 3 = 1

3 18 + 36 ÷ 3 − 3 × 5 × 2 − 3 × 1 = 9



Theoretical probability for

single-step experimentsThere are many situations where the probabilities of experiments can be calculated theoretically. This means we can calculate the probabilities without actually doing an experiment. Finding probabilities when tossing a coin or a die, using a spinner, picking a card from a pack of cards or a coloured marble out of a bag are all such examples.

We can list the sample space that gives the total number of outcomes, then count the number of successful outcomes. This allows us to calculate the required probability using:

Pr(event) =

Worked Example 14Alvin has a normal pack of 52 playing cards and he selects one card from the pack.

(a) List the sample space.

(b) Find the probability of that card being:

(i) a diamond (ii) a king or a queen.

Thinking(a) 1 List the sample space. (a) ♥: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K

♦: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K♣ : A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K♠ : A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K

(♥ = Hearts, ♦ = Diamonds, ♣ = Clubs, ♠ = Spades, A = Ace, J = Jack, Q = Queen, K = King)

(b) (i) 1 Count the number of successful outcomes in the sample space.Count the total number of outcomes.

(b) (i) Number of successful outcomes = 13

Total number of outcomes = 52

2 Calculate the probability by using

Pr(event) =

Pr(diamond) =

=


------------------------------------------------------------------------------- .

14


-------------------------------------------------------------------------------

1352------

14---

9.6AA Pearson Maths 8 SB-09.fm Page 562 Wednesday, August 10, 2011 2:26 PM

Documents

Pearson Mathematics 8 SB COVER - Weebly