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WJEC MATHEMATICS

INTERMEDIATE

STATISTICS AND PROBABILITY

PROBABILITY AND TREE

DIAGRAMS

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Contents

All Probabilities are Between 0 and 1

Probabilities Add up to 1

Listing All Outcomes

Expected Probability

The AND / OR Rule

Tree Diagrams

Probability from Venn Diagrams

Credits

Probability scale

https://sites.google.com/a/egrps.org/murphys-math/probability-1

WJEC Question bank

http://www.wjec.co.uk/question-bank/question-search.html

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All Probabilities are between 0 and 1

Probabilities are always between 0 and 1. The higher the probability of

something, the more likely it is to happen

A probability of 0 means it NEVER WILL happen

A probability of 1 means it DEFINITELY WILL happen

Probability formula

Example

Calculate the probability of selecting a vowel from the tiles below

Probability =

Probability = Number of ways for something to happen

Total number of possible results

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Probabilities Add up to 1

There are two key facts we need to know:

1. If only one result can happen at a time, then all the probabilities will add

up to one

2. Since something must either happen or not happen;

The probability it happens + The probability it doesn't happen = 1

Example 1

Example 2

The probability John is late for work is 0.26. What is the probability he is not

late for work

From the second key fact

1 - 0.26 = 0.74

Exercise S4

1. Complete the following tables

Colour Red Green Blue

Probability 0.52 0.3

Transport Bike Car Train Walk Plane Other

Probability 0.24 0.41 0.16 0.14 0.03

Card Card 1 Card 2 Card 3 Card 4

Probability 0.45 0.15 0.09

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2.

One of the following shapes are chosen at random

What is the probability of that shape being a triangle?

What is the probability of that shape being a square?

What is the probability of that shape being a circle?

What is the probability of that shape not being a circle?

3. Jamie selects a number between (and including) 1 and 25.

a) What is the probability that his selected number is a multiple of 6?

b) What is the probability that his selected number is a multiple of 5?

c) What is the probability that his selected number is a square number?

d) What is the probability that his selected number is prime?

e) What is the probability that his selected number is not a 1 digit

number?

4. Alice selects a letter a random from the word BANANA.

a) What is the probability that her selected letter is the letter N?

b) What is the probability that her selected letter is a vowel?

c) What is the probability that her selected letter is a T?

5. The probability Jasmine drives to work is 0.73. Calculate the probability

that she does not drive to work

6. The probability of it being sunny tomorrow is

. What is the probability

that it will not be sunny?

7. When rolling a fair 20 sided die, what is the probability of not rolling a factor

of 24?

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Listing All Outcomes

Listing all outcomes, is just listing all the things that could happen. Often, we

are asked to create a sample space diagram.

A sample space diagram is a good way to show all the possible outcomes if

there are two activities going on (e.g. two coins being thrown, two dice being

thrown, or two spinners).

Example

The following two spinners are spun and the numbers on both are multiplied

together. Create a sample space diagram to show all possible outcomes.

1 2 4 6 8

3 6 12 18 24

5 10 20 30 40

2 4 6 8

The number of values in the table (highlighted) gives us the total number of

possible results. This is very useful for using the probability formula. For

example, you may be asked to calculate the probability of your score being 6

Probability =

Spinner 1

Spinner 2

There are 12 items in the table so

there are 12 possible outcomes

There are 2 sixes in the table. i.e.

there are 2 ways of getting a 6

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Expected Frequency

Example 2

The probability of winning a game is

. If a player plays the game 180 times,

how many would you expect them to win?

Using the above formula:

Expected number of wins =

x 180 = 60

Exercise S5

1.

Expected times something will happen = probability x number of trials

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2.

3.

9

4.

5.

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Relative Frequency

Some probabilities we know (an example would be the probability of flipping a

coin and it landing on heads)

If we don't know the probability, we can calculate an estimate of it through

repeated experiment. In this case, instead of using the word 'Probability' we

use "Relative Frequency"

The following table shows results of 100 rolls of an untested die.

Score 1 2 3 4 5 6

Frequency 3 20 50 7 15 5

(a) What is the relative frequency of obtaining a 3

(b) What is the probability of scoring 5 or more

(c) If die is rolled 600 times, how many times would you expect

to get a 1

From above, it shows you get

3 1s in 100 rolls, so will get 3 x 6 = 18 1s in 100 throws

Note: From above, it seems as if the dice is unfair as you would

expect approximately 17 for each value (100 6)

MORE ROLLS (TRIALS) WOULD MAKE THE RESULTS MORE

RELIABLE

Relative Frequency =

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Exam Questions S7

1.

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2.

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And / Or Rule

If you are ask to find the probability of event A and event B, you multiply the

probabilities together

If you are asked to find the probability of event A or event B, you add the

probabilities together

Example 1

A bag contains 5 red balls, 4 yellow balls, and 3 green balls. One ball is

randomly selected from the bag. Find the probability that the selected ball is

red or yellow.

P(red) =

P(yellow) =

P(red or yellow) =

Example 2

The probability that Jane wears a dress to work is 0.3. The probability that

she walks to work is 0.2. Find the probability that Jane wears a dress and

she walked

P(dress) = 0.12 P(walk) = 0.2

P(dress and walk) = 0.3 x 0.2 = 0.6

Example 3

The probability Jane wears a hat is 0.3. The probability she wears a hat and

eats a burger is 0.12. Find the probabiliy she eats a burger

0.3 x P(Burger) = 0.12 P(Burger)= 0.4

AND OR

x +

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Tree Diagrams

A tree diagram is a way of seeing all the possible probability 'routes' for two

(or more) events. A game consists of selecting a counter from a bag

(containing three red counters and seven blue) twice.

Question 1: Find the probability that a player selects two red counters.

(This path has been drawn on the tree diagram with arrows.)

Answer:

Question 2: Find the probability the two counters are different colours

P(Red and Blue) =

P(Blue or Red) =

P(Red and Blue OR Blue and Red) =

Important

Each set of lines

that meet at the

same point

MUST add to 1

Important

When travelling along branches, you MULTIPLY

This is and rule

as we need red

and red.

This means we need:

red and blue OR blue and red

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Exam Questions S8

1.

2.

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3.

4.

17

5.

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Probability from Venn Diagrams

80 pupils in a certain school may choose one, two or three optional

subjects

History (H), Geography (G) and French (F).

The numbers in the Venn diagram represent the number of pupils in

each subset.

If a pupil is chosen at random from the group, find the probability that

(c) he studies Geography,

(d) he studies one optional subject only.

If it is known a pupil studies History, find the probability that

(g) he studies biology as well.

(h) he studies geography but not biology.

G

H

16

20

5

1 3 7

8

21

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B