GCSE Statistics · Higher · Chapter 11: Probability · Lesson 4

Conditional probability

About 55 minutesCalculator allowed40 marks

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Date

Do Now

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
A bag has 6 blue pens and 3 black pens. Two are taken out at random, one after the other, and not put back. Find the probability that they are the same colour.
[2 marks]
Answer
A2
From chapter 8
Over four years the price of a litre of petrol rose by $11%$ , rose by $9%$ , rose by $6%$ and then fell by $2%$ . Work out the average percentage increase per year, to 1 decimal place.
[2 marks]
Answer
A3
From chapter 11
The probability that a customer at a café buys a drink is $0.35$ . The probability that the customer buys a cake is $0.6$ , and the probability that the customer does both is $0.1$ . Find the probability that the customer buys a drink or a cake or both.
[2 marks]
Answer
A4
Maths skill
Amy and Ben share some sweets in the ratio $3 : 5$ . Ben gets $8$ more sweets than Amy. How many sweets are there altogether?
[2 marks]
Answer

Example, then your turn

Your teacher works through each example with you. Then try the one beside it.

Example 1

$P (A) = 0.5$ , $P (B) = 0.4$ and $P (A \cap B) = 0.2$ . Work out $P (A ∣ B)$ . Are $A$ and $B$ independent?

Hint

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}

. Independent events have

P (A ∣ B) = P (A)

B3
$P (X) = 0.6$ , $P (Y) = 0.35$ and $P (X \cap Y) = 0.24$ . Are $X$ and $Y$ independent? You must show your working.
[2 marks]

Example 2

The table shows how 150 students have lunch. (a) Find the probability that a Year 11 student, picked at random, brings a packed lunch. (b) Find the probability that a student with a packed lunch, picked at random, is in Year 11.

Hint The condition tells you which group you choose from: that group's total is the denominator.

B1
The table shows the main activity of 200 people at a sports club.
(a)
Find the probability that a female member, picked at random, mainly swims.
[1 mark]
Answer
(b)
Find the probability that a member who mainly swims, picked at random, is female.
[2 marks]
Answer
Total for B1: 3 marks

Example 3

The Venn diagram shows 80 people and whether they drink tea ( $T$ ) or coffee ( $C$ ). Work out (a) $P (C ∣ T)$ , (b) $P (C ∣ T^{'})$ .

Hint

T^{'}

is everyone outside circle

T

, including the people outside both circles.

B2
The Venn diagram gives $P$ for each region, for events $A$ and $B$ .
(a)
Work out $P (A ∣ B)$ .
[1 mark]
Answer
(b)
Work out $P (B ∣ A^{'})$ .
[2 marks]
Answer
Total for B2: 3 marks

Example 4

A factory has two machines. Machine $A$ makes $70%$ of the parts and machine $B$ makes $30%$ . $2%$ of the parts from $A$ and $5%$ of the parts from $B$ are faulty. A part is faulty. Work out the probability that it was made by machine $B$ .

Hint Work out

P (faulty)

first: it is the denominator.

B4
A shop sends $22%$ of its parcels by van and the rest by post. $9%$ of the van parcels and $4%$ of the posted parcels arrive late.
(a)
Work out the probability that a parcel arrives late.
[2 marks]
Answer
(b)
A parcel arrives late. Work out the probability that it was sent by van.
[2 marks]
Answer
Total for B4: 4 marks

Practice

C1
For two events, $P (A) = 0.35$ and $P (A \cap B) = 0.14$ . Work out $P (B ∣ A)$ .
[2 marks]
Answer
C2
The probability that a match is played in rain is $0.3$ . Given rain, the probability that the match is called off is $0.7$ . Work out the probability that a match is played in rain and is called off.
[2 marks]
Answer

C3
Of 200 students, 120 walk to school, 50 have a dog and 30 walk to school and have a dog. Are walking to school and having a dog independent? Show how you decide.
[2 marks]
C4
The table shows 1000 men. Liam says, "The probability that a man with heart disease smokes is $\frac{30}{200}$ ."
(a)
What probability has Liam worked out?
[1 mark]
(b)
Work out the probability that a man with heart disease smokes.
[2 marks]
Answer
Total for C4: 3 marks

Exam-style questions

D1
For two events, $P (A) = 0.3$ and $P (B) = 0.4$ . $A$ and $B$ are mutually exclusive. Sam says, "Mutually exclusive events are independent."
Use these events to show that Sam is wrong.
[2 marks]

D2
A survey of 200 households found that 46 have a dog but no garden, 64 have a dog and a garden, 38 have a garden but no dog and 52 have neither.
(a)
Find the probability that a household with a dog, picked at random, has a garden.
[2 marks]
Answer
(b)
Compare the probability of having a dog for households with a garden and households without a garden.
[2 marks]
(c)
Are having a dog and having a garden independent? Give a reason.
[2 marks]
Total for D2: 6 marks

Extension

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A bag has 4 red and 6 blue counters. Ella takes two of them at random, one after the other, and does not put the first back. At least one of her counters is red. Work out the probability that both are red.
[3 marks]
Hint 1 · What to try
The condition "at least one red" is the denominator.
Hint 2 · The first line
$P (both red) = \frac{4}{10} \times \frac{3}{9} = \frac{12}{90}$ and $P (at least one red) = 1 - \frac{6}{10} \times \frac{5}{9} = \frac{60}{90}$ .
Hint 3 · The full method
$\frac{12}{90} \div \frac{60}{90} = \frac{12}{60} = \frac{1}{5}$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out a conditional probability from a table, a Venn diagram or a tree, using the group in the condition as the denominator.
I can use $P (B ∣ A) = \frac{P ( A \cap B )}{P ( A )}$ , forwards and backwards.
I can test whether two events are independent and give a conclusion.

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