© boardworks ltd 2004 1 of 55 d2d2 d2d2 d2d2 d2d2 d2d2 d4.1 the language of probability contents d4...

© Boardworks Ltd 2004 1 of 55

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D4.1 The language of probability

Contents

D4 Probability

D4.5 Experimental probability

D4.2 The probability scale

D4.4 Probability diagrams

D4.3 Calculating probability


The language of probability

Probability is a measurement of the chance or likelihood of an event happening.

Probability is a measurement of the chance or likelihood of an event happening.

Describe the chance of drawing a red marble.

Unlikelyمرجح غير

Likelyمرجح

Certainمؤكد

Impossibleمستحيل

even Chanceمتساوي الفرصة


The probability scale

The chance of an event happening can be shown on a probability scale.

impossible certaineven chanceunlikely likely

Less likely More likely

Meeting with King

Henry VIII

A day of the week starting

with a T

The next baby born being a

boy

Getting a number > 2 when roll a

fair dice

A square having four right angles


Fair games

A game is played with marbles in a bag.

One of the following bags is chosen for the game. The teacher then pulls a marble at random from the chosen bag:

If a red marble is pulled out of the bag, the girls get a point.

If a blue marble is pulled out of the bag, the boys get a point.

Which would be the fair bag to use?

bag abag a bag cbag cbag bbag bbag bbag b


Fair games

A game is fair if all the players have an equal chance of winning.

A game is fair if all the players have an equal chance of winning.

Which of the following games are fair?

A dice is thrown. If it lands on a prime number team A gets a point, if it doesn’t team B gets a point.

There are three prime numbers (2, 3 and 5) and three non-prime numbers (1, 4 and 6).

Yes, this game is fair.


Fair games

Nine cards numbered 1 to 9 are used and a card is drawn at random. If a multiple of 3 is drawn team A gets a point.If a square number is drawn team B gets a point.If any other number is drawn team C gets a point.

There are three multiples of 3 (3, 6 and 9).

No, this game is not fair. Team C is more likely to win.

There are three square numbers (1, 4 and 9).

There are four numbers that are neither square nor multiples of 3 (2, 5, 7 and 8).


Fair games

A spinner has five equal sectors numbered 1 to 5.The spinner is spun many times.If the spinner stops on an evennumber team A gets 3 points.If the spinner stops on an odd number team B gets 2 points.

1

23

4

5

Suppose the spinner is spun 50 times.We would expect the spinner to stop on an even number 20 times and on an odd number 30 times.Team A would score 20 × 3 points = 60 pointsTeam B would score 30 × 2 points = 60 points

Yes, this game is fair.


Bags of counters

You are only allowed to choose one counter at random from one of the bags.

Which of the bags is most likely to win a prize?

Choose a blue counter and win a prize!

bag abag a bag bbag b bag cbag cbag cbag c



The chance of an event happening can be shown on a probability scale.

impossible certaineven chanceunlikely likely

Less likely More likely

Meeting with King

Henry VIII

A day of the week starting

with a T

The next baby born being a

boy

Getting a number > 2 when roll a

fair dice

A square having four right angles



We measure probability on a scale from 0 to 1.

If an event is impossible or has no probability of occurring then it has a probability of 0.

If an event is certain it has a probability of 1.

This can be shown on the probability scale as:

Probabilities are written as fractions, decimal and, less often, as percentages between 0 and 1.

0 ½ 1impossible certaineven chance


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Contents

D4 Probability






Higher or lower


Listing possible outcomes

When you roll a fair dice you are equally likely to get one of six possible outcomes:

16

16

16

16

16

16

Since each number on the dice is equally likely the probability of getting any one of the numbers is 1 divided by 6 or .1

6


Calculating probability

What is the probability of the following events?

P(tails) = 12

P(red) = 14

P(7 of ) = 152

P(Friday) = 17

2) This spinner stopping on the red section?

3) Drawing a seven of hearts from a pack of 52 cards?

4) A baby being born on a Friday?

1) A coin landing tails up?



If the outcomes of an event are equally likely then we can calculate the probability using the formula:

Probability of an event =Number of successful outcomes

Total number of possible outcomes

For example, a bag contains 1 yellow, 3 green, 4 blue and 2 red marbles.

What is the probability of pulling a green marble from the bag without looking?

P(green) =310

or 0.3 or 30%



This spinner has 8 equal divisions:

a) a red sector?b) a blue sector?c) a green sector?

What is the probability of the spinner landing on

a) P(red) =28

=14

b) P(blue) =18

c) P(green) =48

=12



A fair dice is thrown. What is the probability of gettinga) a 2?b) a multiple of 3?c) an odd number?d) a prime number?e) a number bigger than 6?f) an integer?

a) P(2) = 16

b) P(a multiple of 3) = 26

=13

c) P(an odd number) = 36

=12



A fair dice is thrown. What is the probability of gettinga) a 2?b) a multiple of 3?c) an odd number?d) a prime number?e) a number bigger than 6?f) an integer?

d) P(a prime number) = 36

e) P(a number bigger than 6) =

f) P(an integer) = 66

= 1

=12

0

Don’t write 0

6


Calculating probabilities

Answer these questions giving each answer as a fraction or 0 or 1.


The probability of an event not occurring

If the probability of an event occurring is p then the probability of it not occurring is 1 – p.If the probability of an event occurring is p then the probability of it not occurring is 1 – p.

The following spinner is spun once:

What is the probability of it landing on the yellow sector?

P(yellow) =14

What is the probability of it not landing on the yellow sector?

P(not yellow) =34



The probability of a factory component being faulty is 0.03. What is the probability of a randomly chosen component not being faulty?

P(not faulty) = 1 – 0.03 = 0.97

The probability of pulling a picture card out of a full deck of

cards is .

What is the probability of not pulling out a picture card?

3

13

P(not a picture card) = 1 – =313

1013



The following table shows the probabilities of 4 events. For each one work out the probability of the event not occurring.

EventProbability of the event occurring

Probability of the event not occurring

A

B

C

D

3

5

9

20

0.77

8%

2

5

11

20

0.23

92%



There are 60 sweets in a bag.

What is the probability that a sweet chosen at random from the bag is:

a) Not a cola bottle56

P(not a cola bottle) =

b) Not a teddy4560

P(not a teddy) =

10 are cola bottles, 14

are fried eggs,

the rest are teddies.20 are hearts,

=34


Adding mutually exclusive outcomes

If two outcomes are mutually exclusive then their probabilities can be added together to find their combined probability.

If two outcomes are mutually exclusive then their probabilities can be added together to find their combined probability.

What is the probability that a card is a moon or a sun?

P(moon) =13

and P(sun) =13

Drawing a moon and drawing a sun are mutually exclusive outcomes so,P(moon or sun) = P(moon) + P(sun) =

13

+13

= 23

For example, a game is played with the following cards:


Adding mutually exclusive outcomes

What is the probability that a card is yellow or a star?

P(yellow card) =13

and P(star) =13

Drawing a yellow card and drawing a star are not mutually exclusive outcomes because a card could be yellow and a star.

P (yellow card or star) cannot be found simply by adding.

P(yellow card or star) =

We have to subtract the probability of getting a yellow star.

13

+13

–19

=3 + 3 – 1

9=

59


The sum of all mutually exclusive outcomes

The sum of all mutually exclusive outcomes is 1.The sum of all mutually exclusive outcomes is 1.

For example, a bag contains red counters, blue counters, yellow counters and green counters.

P(blue) = 0.15 P(yellow) = 0.4 P(green) = 0.35

What is the probability of drawing a red counter from the bag?

P(blue or yellow or green) = 0.15 + 0.4 + 0.35 = 0.9

P(red) = 1 – 0.9 = 0.1


Finding all possible outcomes of two events

Two coins are thrown. What is the probability of getting two heads?

Before we can work out the probability of getting two heads we need to work out the total number of equally likely outcomes.

There are three ways to do this:

1) We can list them systematically.

Using H for heads and T for tails, the possible outcomes are:

TH and HT are separate equally likely outcomes.HH.TT, TH, HT,



2) We can use a two-way table.

Second coin

H T

H

TFirstcoin

HH HT

TH TT

From the table we see that there are four possible outcomes one of which is two heads so,

P(HH) =14



3) We can use a probability tree diagram.

First coinH

T

Second coinH

TH

T

Outcomes

HH

HTTH

TT

Again we see that there are four possible outcomes so,

P(HH) =14


Finding the sample space

A red dice and a blue dice are thrown and their scores are added together.

What is the probability of getting a total of 8 from both dice?

There are several ways to get a total of 8 by adding the scores from two dice.

We could get a 2 and a 6, a 3 and a 5, a 4 and a 4,a 5 and a 3, or a 6 and a 2.

To find the set of all possible outcomes, the sample space, we can use a two-way table.


Finding the sample space

+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

From the sample space we can see that there are 36 possible outcomes when two dice are thrown.

Five of these have a total of 8.

8

8

8

8

8P(8) =

536


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Contents

D4 Probability






Estimating probabilities based on data

Suppose 1000 people were asked whether they were left- or right-handed.

Of the 1000 people asked 87 said that they were left-handed.

If we repeated the survey with a different sample the results would probably be slightly different.

From this we can estimate the probability of someone being

left-handed as or 0.087.87

1000

The more people we asked, however, the more accurate our estimate of the probability would be.


Relative frequency

The probability of an event based on data from an experiment or survey is called the relative frequency.

The probability of an event based on data from an experiment or survey is called the relative frequency.

Relative frequency is calculated using the formula:

Relative frequency =Number of successful trials

Total number of trials

For example, Ben wants to estimate the probability that a piece of toast will land butter-side-down.

He drops a piece of toast 100 times and observes that it lands butter-side-down 65 times.

Relative frequency =65100

=1320


Relative frequency

Sita wants to know if her dice is fair. She throws it 200 times and records her results in a table:

Number Frequency Relative frequency

1 31

2 27

3 38

4 30

5 42

6 32

Is the dice fair?

312002720038200302004220032200

= 0.155

= 0.135

= 0.190

= 0.150

= 0.210

= 0.160


Experimental probability


Expected frequency

The theoretical probability of an event is its calculated probability based on equally likely outcomes.

Expected frequency = theoretical probability × number of trialsExpected frequency = theoretical probability × number of trials

If you rolled a dice 300 times, how many times would you expect to get a 5?

The theoretical probability of getting a 5 is .16

So, expected frequency = × 300 = 16

50

If the theoretical probability of an event can be calculated, then when we do an experiment we can work out the expected frequency.


Expected frequency

If you tossed a coin 250 times how many times would you expect to get a tail?

Expected frequency = × 250 = 12

125

If you rolled a fair dice 150 times how many times would you expect

to a number greater than 2?

Expected frequency = × 150 = 23

100


Spinners experiment

* Write the sample space ( all possible results ) when rolling a fair dice

Worksheet ( 1 )

نرد ( ) حجر إلقاء عند الممكنة النواتج جميع مجموعة العينة فضاء اكتبمنتظم

Coin 2

H T

Coin 1

H

T

1) Complete the table to show all possible results .

Use the words : impossible , unlikely , even chance , likely , certain to describe the following events :

1) The upper face is a number greater than 5. ……………….

2) The upper face is a prime number. ……………….

* You toss 2 coins together

1) You will get 2 heads ……………….

2) At least one head ……………….


3) You will get one tail exactly……………….

المصطلحات : استخدم ، مرجح غير ، مستحيل

، مرجح ، الفرصة متساويالتالية األحداث لتصف : مؤكد

المصطلحات : استخدممتساوي ، مرجح غير ، مستحيللتصف مؤكد ، مرجح ، الفرصة

التالية : األحداث

من اكبر عدد العلوي 5ظهور الوجه على

العلوي الوجه على أولي عدد ظهور

K معا نقود قطعتي ألقيت

صورتين على ستحصل

األقل على واحدة صورة ستظهر

بالضبط واحدة مرة الكتابة ستظهر

النواتج جميع لتبين الجدول أكملالممكنة

2 cards were randomly drawn from a deck of 52 cards

Worksheet ( 2 )

تحوي ( ) التي الشدة الورق لعب علبة من ورقتين سحب تمورقة 52

Card 2

Card 1

: Spade بستوني : Clubs سباتي : Diamond ديناري : Heart ( ) قلب كبه

1) Complete the table to show all possible results .


1) The 2 cards are of the same color. ……………….

2) The 2 cards are spades. ……………….

3) One of the cards was green. ……………….

4) The 2 cards are either red or black or red and black……………….

5) At least one of the 2 cards wasn’t a picture. ……………….

النواتج جميع لتبين الجدول أكملالممكنة

المصطلحات : استخدم ، مرجح غير ، مستحيل

، مرجح ، الفرصة متساويالتالية األحداث لتصف : مؤكد

اللون نفس من البطاقتان

البستوني نوع من البطاقتان

خضراء البطاقتين احد

حمراء أو سوداوتان أو حمراوتان إما البطافتانوسوداء

صورة ليست احدهما األقل على

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