Conditional probability.pptx

EDEXCEL GCSE MATHS · HIGHER

Conditional probability

Probability

Lesson 5 of 6

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EDEXCEL GCSE MATHS · HIGHER

Warm-up

Probability

Lesson 5 of 6

Answer each one, then check.

1

Work out ⅝ × 4/7.

2

Simplify 30/56.

3

Work out 1 − 5/14.

4

What does "independent" mean?

5

What does "with replacement" mean?

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EDEXCEL GCSE MATHS · HIGHER

Warm-up - Answers

Probability

Lesson 5 of 6

1

Work out ⅝ × 4/7.

5/14

2

Simplify 30/56.

15/28

3

Work out 1 − 5/14.

9/14

4

What does "independent" mean?

One event does not affect the other

5

What does "with replacement" mean?

The item is put back before the next pick

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EDEXCEL GCSE MATHS · HIGHER

Learning Objectives

Probability

Lesson 5 of 6

1

Explain what conditional probability means.

2

Draw a tree diagram for events without replacement.

3

Work out probabilities of combined events without replacement.

4

Find a conditional probability from a two-way table.

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EDEXCEL GCSE MATHS · HIGHER

Probability

Lesson 5 of 6

CONDITIONAL PROBABILITY

The probability of an event happening, given that another event has already happened, is a conditional probability.

Without replacement, the numbers change after each pick.

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EDEXCEL GCSE MATHS · HIGHER

A Tree Without Replacement

Probability

Lesson 5 of 6

After one counter is taken, only 7 are left, so the second-stage probabilities change.

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EDEXCEL GCSE MATHS · HIGHER

Without Replacement

Probability

Lesson 5 of 6

A bag has 5 red and 3 blue counters. Two counters are taken at random without replacement. Find the probability that both are red.

1

First counter red

⅝

2

Now 4 red among 7 remaining

4/7

3

Multiply

⅝ × 4/7 = 20/56

4

Simplify

5/14

ANSWER

5/14

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EDEXCEL GCSE MATHS · HIGHER

One of Each Colour

Probability

Lesson 5 of 6

For the same bag, find the probability that the two counters are different colours.

1

Red then blue

⅝ × 3/7 = 15/56

2

Blue then red

⅜ × 5/7 = 15/56

3

Add

30/56

4

Simplify

15/28

ANSWER

15/28

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EDEXCEL GCSE MATHS · HIGHER

At Least One Red

Probability

Lesson 5 of 6

A bag has 4 red and 6 blue counters. Two are taken without replacement. Find the probability that at least one is red.

1

At least one red is the opposite of both blue

Use 1 − P(both blue)

2

Both blue

6/10 × 5/9 = 30/90 = ⅓

3

Subtract from 1

1 − ⅓ = ⅔

ANSWER

⅔

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EDEXCEL GCSE MATHS · HIGHER

Two-Way Tables

Probability

Lesson 5 of 6

Conditional probabilities can be read from a table by restricting to a group.

The idea

For "given that", use only the people in that group as the total.

Example

60 students: 35 like football, and 20 of those are boys. Given that a student likes football, the probability that they are a boy is 20/35.

Simplify

20/35 = 4/7.

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EDEXCEL GCSE MATHS · HIGHER

A Two-Way Table

Probability

Lesson 5 of 6

60 students were asked whether they like football.

Likes football

Does not like football

Total

Boys

20

10

30

Girls

15

15

30

Total

35

25

60

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EDEXCEL GCSE MATHS · HIGHER

Given That

Probability

Lesson 5 of 6

Use the table. A student is chosen at random. Find the probability that they are a boy given that they like football.

1

Restrict to students who like football

35 students

2

Boys among them

20

3

Probability

20/35 = 4/7

ANSWER

4/7

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EDEXCEL GCSE MATHS · HIGHER

Key Terms

Probability

Lesson 5 of 6

Conditional probability

The probability of an event given that another has happened.

Without replacement

The item is not put back, so later probabilities change.

Dependent events

Events where one affects the probability of the other.

Tree diagram

A diagram showing outcomes and probabilities on branches.

Two-way table

A table showing two categories at once.

Complement

The event "not A".

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EDEXCEL GCSE MATHS · HIGHER

Probability

Lesson 5 of 6

YOUR TASK

Cards Without Replacement

12 minutes

Three cards are taken from a pack of ten cards numbered 1 to 10, one after another without replacement. Find the probability that the first is even and the second is odd, and that the first two are both greater than 7.

1

Reduce the numbers after each pick.

2

Multiply along the path.

WHAT A GOOD ANSWER SHOWS

First even and second odd: 5/10 × 5/9 = 25/90 = 5/18. Both greater than 7: 3/10 × 2/9 = 6/90 = 1/15.

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EDEXCEL GCSE MATHS · HIGHER

Can I...?

Probability

Lesson 5 of 6

Explain what conditional means.

Adjust the numbers after a pick.

Draw a tree diagram without replacement.

Multiply along the branches.

Add paths for combined outcomes.

Use "1 minus" for at least one.

Find a conditional probability from a table.

Simplify my fractions.

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EDEXCEL GCSE MATHS · HIGHER

Summary & Exam Focus

Without replacement, probabilities on the second branches change.

Multiply along the branches; add different paths.

"Given that" means restrict to that group.

At least one = 1 − none.

EXAM FOCUS

A bag has 5 red and 3 blue counters. Two counters are taken at random without replacement. Work out the probability that the counters are different colours. (3 marks)

Both orders count: red then blue AND blue then red. Reduce the total by 1 for the second pick.

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