GCSE Maths · Higher · Chapter 19: Combined events · Lesson 1

Combining two events

Grades 5–7About 50 minutesNo calculator41 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
Two fair dice are rolled. Find P(total is 11).
[1 mark]
Answer
A2
From chapter 3
What type of correlation would you expect between height and arm span?
[1 mark]
Answer
A3
From chapter 15
$x^{3} = 50$ . $3. 6^{3} = 46.66$ , $3. 7^{3} = 50.65$ and $3.6 5^{3} = 48.63$ . Give $x$ to 1 decimal place.
[1 mark]
Answer
A4
From chapter 17
Write down the turning point of $y = (x - 3)^{2} + 1$ .
[1 mark]
Answer

Grades 5–6

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

Example 1

A spinner lands on red, blue, green or yellow. $P (red) = 0.26$ and $P (blue) = 0.33$ . Work out the probability that it lands on green or yellow.

Hint It can only land on one colour, so the four probabilities add to

1

. Green or yellow is everything that is not red or blue.

B1
A bag holds red, white, black and yellow counters. A counter is taken at random. $P (red) = 0.18$ , $P (white) = 0.27$ and $P (black) = 0.4$ . Work out the probability that it is red or yellow.
[2 marks]
Answer

Example 2

$P (A) = 0.5$ , $P (B) = 0.35$ and $P (A and B) = 0.2$ . Work out $P (A or B)$ .

Hint Adding

P (A)

and

P (B)

counts the overlap twice, so take it off once.

B2
$P (A) = 0.42$ , $P (B) = 0.36$ and $P (A and B) = 0.18$ . Work out the probability that neither $A$ nor $B$ happens.
[2 marks]
Answer

Grades 5–6

C1
A fair dice is thrown. Work out the probability that the score is even or greater than $4$ .
[2 marks]
Answer
C2
$P (A) = 0.65$ and $P (B) = 0.5$ . The probability that neither $A$ nor $B$ happens is $0.15$ . Work out $P (A and B)$ .
[3 marks]
Answer

C3
A spinner lands on red, blue, green or yellow. The table shows some of the probabilities. Work out the probability that it lands on green or yellow.
[3 marks]
Answer
C4
The Venn diagram shows probabilities. Work out $P (B)$ .
[3 marks]
Answer
C5
A card is taken at random from $24$ cards numbered $1$ to $24$ . Explain why "a multiple of $4$ " and "a multiple of $6$ " are not mutually exclusive. Then work out the probability that the card is a multiple of $4$ or a multiple of $6$ .
[3 marks]
Answer
C6
Spinner $P$ is numbered $1$ , $2$ and $3$ . Spinner $Q$ is numbered $1$ to $5$ . Both are fair and both are spun. List the outcomes, then work out the probability that the two numbers add up to $6$ .
[2 marks]
Answer

Grades 6–7

D1
A card is taken at random from an ordinary pack of $52$ playing cards. Noah says, "The probability that it is a king or a heart is $\frac{4}{52} + \frac{13}{52} = \frac{17}{52}$ ." Explain what Noah has done wrong, and work out the correct probability. Give your answer in its simplest form.
[3 marks]
Answer

D2
In a game, spinner $P$ (numbered $1$ , $2$ and $3$ ) and spinner $Q$ (numbered $1$ to $5$ ) are both spun and the two numbers are multiplied. Kai wins if the answer is odd. Lily wins if it is even. Kai says, "The game is fair, because odd and even numbers are equally likely." Is Kai right? You must show your working.
[4 marks]
D3
There are $80$ students in a year group. $34$ of them play an instrument and $25$ sing in the choir. $11$ do both. In the Venn diagram, $I$ is the students who play an instrument and $C$ is the students in the choir.
(a)
Complete the Venn diagram. How many of the students do neither?
[2 marks]
Answer
(b)
A student is chosen at random. Work out the probability that the student plays an instrument or sings in the choir. Give your answer in its simplest form.
[2 marks]
Answer
(c)
A student is chosen at random. Work out the probability that the student does exactly one of the two.
[2 marks]
Answer
Total for D3: 6 marks

Grade 8 and beyond

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can add the probabilities of outcomes that cannot happen together.
I can use $P (A or B) = P (A) + P (B) - P (A and B)$ .
I can list a sample space and count from it.

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