GCSE Maths · Higher · Chapter 19: Combined events · Lesson 3

Conditional probability

Grades 6–8About 50 minutesNo calculator38 marks

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NameClassDate

Do Now

Grades 4–6

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

A1
Needed today
Independent events: P(A) $= 0.4$ , P(B) $= 0.5$ . Find P(A and B).
[1 mark]
Answer
A2
From chapter 18
A class of width 10 has frequency 4. Find its frequency density.
[1 mark]
Answer
A3
From chapter 11
A right-angled triangle has shorter sides 7 cm and 24 cm. Find the hypotenuse.
[1 mark]
Answer
A4
From chapter 4
Find the next term: $3, 6, 12, 24, \dots$
[1 mark]
Answer

Example, then your turn

Grades 6–7

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

Example 1

The table shows the snacks bought by some cinema customers. Given that a customer is a child, work out the probability that they bought popcorn.

Hint "Given that the customer is a child" means only the child row counts. There are

27 + 9 + 9 = 45

children.

B1
The table shows the tickets bought by some train passengers. Given that a passenger travels in the evening, work out the probability that they have a return ticket.
[2 marks]
Answer

Example 2

The tree diagram shows the probability that Lucy's train is late, and the probability that Lucy is late for work. Given that Lucy is late, work out the probability that the train was late.

Hint Lucy is late on two paths:

0.2 \times 0.6 = 0.12

and

0.8 \times 0.05 = 0.04

. Only the first has a late train.

B2
The tree diagram shows the probability that a bulb is made on a factory's night shift, and the probability that it is faulty. A faulty bulb is picked at random. Work out the probability that it was made at night.
[3 marks]
Answer

Practice

Grades 6–7

C3
Two fair dice are thrown. Given that the total score is $8$ , work out the probability that one of the dice shows a $6$ .
[2 marks]
Answer
C4
A club has $75$ members, $40$ of them adults. An adult swims with probability $0.35$ , and $20$ children swim. A swimmer is picked at random. Work out $P (adult)$ .
[3 marks]
Answer

C1
The Venn diagram shows how many students take art ( $A$ ) and drama ( $D$ ). A student who takes exactly one of them is picked at random. Work out $P (drama)$ .
[2 marks]
Answer
C2
The Venn diagram shows probabilities. Given that $B$ does NOT happen, work out the probability that $A$ happens.
[2 marks]
Answer
C5
The table shows who sings in a choir. Is being in the choir independent of being a girl? Show how you decide.
[2 marks]
C6
A bag holds $6$ red and $4$ blue counters. Two counters are taken at random without replacement. Given that the second counter is blue, work out the probability that the first was red.
[3 marks]
Answer

Exam-style questions

Grades 7–8

D1
The table shows how some students travel to school. Zara says, "The probability that a Year 10 student cycles is the same as the probability that a cyclist is in Year 10." Show that Zara is wrong.
[3 marks]

D2
A garage tests $250$ cars. $40%$ of them are more than $10$ years old (old). $35%$ of the old cars fail the test. $12%$ of the newer cars fail the test.
(a)
Complete the frequency tree.
[2 marks]
(b)
A car that fails the test is picked at random. Work out the probability that it is old.
[2 marks]
Answer
(c)
One of the $250$ cars is picked at random. Work out the probability that it is old and fails the test.
[1 mark]
Answer
Total for D2: 5 marks
D3
$P (A) = 0.4$ and $P (B) = 0.35$ . The probability that $B$ happens, given that $A$ happens, is $0.3$ . Work out the probability that $A$ happens, given that $B$ happens.
[3 marks]
Answer

Extension

Grade 8 and beyond

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

E1
A bag holds $4$ red, $3$ blue and $2$ green counters. Two counters are taken at random without replacement. Given that the two counters are the same colour, work out the probability that both are red.
[4 marks]
Hint 1 · What to try
First work out the probability that both are red, both blue and both green.
Hint 2 · The first line
$P (both red) = \frac{4}{9} \times \frac{3}{8} = \frac{12}{72}$ . The same colour happens on three of the paths.
Hint 3 · The full method
Divide $P (both red)$ by $P (same colour)$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can restrict to the group "given that" names.
I can find $P (A ∣ B)$ from a tree or a Venn diagram.
I can tell $P (A ∣ B)$ from $P (B ∣ A)$ .

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