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Statistics Chapter 4 Part 2 ::� Conditional Probability

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Chapter Overview

 

1:: Set Notation

 

3:: Formula for Conditional Probability

 

2:: Conditional Probability in Venn Diagrams

Teacher Notes: All of this is from the old S1. The chapter was effectively split into two: all the non-conditional probability content in Year 1 and the rest in Year 2. Set notation was not used in Year 1.

“I have 3 red and 4 green balls in a bag. I take one ball out the bag, keep it, then take another. Given that the second ball was green, determine the probability the first was red.”

4:: Tree Diagrams

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RECAP :: Using sets for sample spaces and events

In general, sets are used to represent collections of items.

 

1

2

3

4

5

6

 

 

 

Each number represents an outcome.

In probability, an event is a set of one or more outcomes. These are the circles in the Venn Diagram.

We use capital letters for the variables representing sets.

 

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Combining events/sets

What does it mean in this context?

​

What is the resulting set of outcomes?

 

 

 

 

 

 

 

 

 

?

 

2

4

6

3

5

1

 

 

 

?

?

?

?

?

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Some fundamentals

What does it mean in this context?

What is the resulting set of outcomes?

 

“A and not B”. Rolling a number which is even and not prime.

 

 

Rolling a number which is not [even or prime].

 

 

Rolling a number which is not [even and prime].

 

 

 

 

 

2

4

6

3

5

1

?

?

?

?

?

?

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?

What area is indicated?

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?

What area is indicated?

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?

What area is indicated?

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?

What area is indicated?

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?

What area is indicated?

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or

alternatively…

 

?

?

What area is indicated?

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?

What area is indicated?

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?

What area is indicated?

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?

What area is indicated?

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Examples

 

Venn Diagram can either contain:

  1. The specific outcomes in each set
  2. The number of items in the set (i.e. frequencies)
  3. The probability of being in that set.

This will usually be stated or made obvious from the context.

 

 

1

12

3

36

 

? Venn Diagram

? a

? b

? c

? d

For union, I visualise this ‘figure-of-8’ shape:

​

​

​

​

Think “A and D”.

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Examples

 

 

 

 

0.05

0.25

0.07

0.08

0.12

0.43

 

 

a

b

c

?

?

?

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Test Your Understanding

The Venn diagram in Figure 1 shows three events A, B and C and the probabilities associated with each region of B. The constants p, q and r each represent probabilities associated with the three separate regions outside B.

 

The events A and B are independent.

 

  1. Find the value of p. (3)

May 2013 (R) Q6

 

?

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Exercise 4E

Pearson Statstics 1

Pages 69-71

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Conditional Probability

Think about how we formed a probability tree at GCSE:

 

 

 

 

 

 

 

 

 

 

Alternatively (and more commonly):

?

?

?

Memory Tip: You’re dividing by the event you’re conditioning on.

Read the ‘|’ symbol as “given that”. i.e. “B occurred given that A occurred”.

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Examples

​

​

​

Total

​

14

38

52

​

26

22

48

Total

40

60

100

 

1

 

a

b

2

Using the Venn Diagram, determine:

 

2

6

4

3

 

 

 

 

?

?

?

?

?

 

a

b

c

?

?

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Further Examples

 

?

 

 

0.4

0.2

?

 

 

0.4

0.1

0.1

0.4

?

a

b

c

Fro Tip: The ‘restricted sample space’ method also works for Venn Diagrams with probabilities.

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Check your understanding

 

 

 

 

0.11

0.17

0.48

0.24

?

?

?

Click to reveal Venn Diagram

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Further Practice (outside of class)

 

1

2

3

?

?

?

?

?

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Further Test Your Understanding

 

May 2013 (R) Q6

 

?

?

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Exercise 4F

Pearson Statistics 1

Pages 72-74

Note: I have skipped Exercise 2B.

 

1

?

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Full Laws of Probability

 

 

 

 

In general:

 

?

?

?

?

?

?

🖉

We first encountered this in the previous section.

 

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Example

 

?

?

?

?

?

Edexcel S1

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Further Examples

 

 

?

?

?

?

 

? a

? b

? c

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Test Your Understanding

a)

 

 

 

 

 

?

?

?

?

Edexcel S1

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SUPER IMPORTANT TIPS

If I were to identify two tips that will possible help you the most in probability questions:

If you see the words ‘given that’, Immediately write out the law for conditional probability.

Example: “Given Bob walks to school, find the probability that he’s not late…”

 

If you see the words ‘are independent’, Immediately write out the laws for independence.

(Even before you’ve finished reading the question!)

 

 

?

?

If you’re stuck on a question where you have to find a probability given others, it’s probably because you’ve failed to take into account that two events are independent or mutually exclusive, or you need to use the conditional probability or additional law.

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Exercise 4H

Pearson Statistics 1

Pages 79-82

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Probability Trees

We saw probability trees in Year 1. The only difference here is determining a conditional probability using your tree.

Example: You have two bags, the first with 5 red balls and 5 blue balls, and the second with 3 red balls and 6 blue balls. You first pick a ball from the first bag, and place it in the second. You then pick a ball from the second bag. Complete the tree diagram.

 

 

 

 

 

 

 

Fro Tip: Use variable subscripting to indicate what pick you’re referring to.

 

 

 

 

 

 

?

?

?

?

?

?

?

It’s vitally important that you use good notation, making use of the | symbol.

?

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Further Example

Edexcel S1 May 2009 Q2

(Part (a) asks for a tree diagram, which may help with this question)

 

?

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Testing Your Understanding

?

?

?

Edexcel S1