GCSE Statistics · Higher · Chapter 11: Probability · Lesson 1

Probability, expected frequency and risk

About 55 minutesCalculator allowed46 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 holds 6 red counters and some blue counters. The probability of taking a red counter is $\frac{2}{3}$ . How many blue counters are in the bag?
[2 marks]
Answer
A2
From chapter 7
Some patients were asked how many times they visited the dentist last year. The answers were: 2 (11 patients), 3 (4 patients), 4 (7 patients), 5 (5 patients). Work out the interquartile range.
[2 marks]
Answer
A3
From chapter 7
A box plot shows the distances, in km, travelled to work by 60 staff: lower quartile 23, median 29, upper quartile 36. Estimate how many of the staff travel less than 23 km.
[2 marks]
Answer
A4
From chapter 9
For a theatre's ticket sales, the first two 4-point moving averages are $119$ and $120$ . The first one starts at Q1 of year 1, when the value was $38$ . Work out the value for Q1 of year 2.
[2 marks]
Answer

Example, then your turn

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

Example 1

A spinner can land on one of four colours. The table shows the probabilities. The spinner is spun 400 times. Work out an estimate for the number of times it lands on green.

Hint The four probabilities add up to

1

B1
A bag holds counters of four colours. The table shows the probability of taking each colour. One counter is taken and replaced, 250 times. Work out an estimate for the number of white counters taken.
[3 marks]
Answer

Example 2

Omar throws a coin and records the total number of heads at four points. (a) Which value gives the best estimate of the probability of heads? Give a reason. (b) Is the coin fair? Give a reason for your answer.

Hint Work out each relative frequency: heads

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throws.

B2
Leah spins a spinner numbered 1 to 5 and records the total number of fives. Work out the best estimate of the probability of a 5, with a reason. Is the spinner fair? Give a reason.
[4 marks]

Example 3

A study looked at people who walk to work and people who drive. (a) Work out the absolute risk of back pain for people who drive. (b) Work out the relative risk of back pain for people who drive compared with people who walk. Interpret your answer. (c) A town has 5000 people who drive to work. Estimate how many of them have back pain.

Hint Absolute risk is the number with back pain divided by the number in that group.

B3
A study looked at cyclists who had an accident.
(a)
Work out the relative risk of a head injury for cyclists with no helmet compared with cyclists who wore a helmet.
[2 marks]
Answer
(b)
Interpret your answer to part (a) in context.
[1 mark]
Total for B3: 3 marks

Example 4

A game costs 60p to play. You roll a fair dice. A six wins £2, a five wins £1 and any other score wins nothing. Is the game fair? You must show your working.

Hint Work out the amount a player wins on average in one game.

B4
A card game costs 80p to play. The table shows the probability of each prize. Is the game fair? You must show your working.
[3 marks]

Practice

C1
The probability that a spinner lands on 4 is $0.15$ . Ali spins it and it lands on 4 sixty times. Estimate the number of times Ali spun it.
[2 marks]
Answer
C2
In a survey, 54 of 240 shoppers paid in cash. Estimate how many of the next 1000 shoppers pay in cash.
[2 marks]
Answer

C3
Amir and Beth each throw the same drawing pin. Amir throws it 20 times and it lands point up 13 times. Beth throws it 200 times and it lands point up 117 times.
(a)
Whose results give the more reliable estimate of the probability of point up? Give a reason.
[1 mark]
(b)
Use all of their results to work out the best estimate of the probability of point up. Give your answer to 2 decimal places.
[2 marks]
Answer
Total for C3: 3 marks
C4
For people under 70, the risk of a reaction to a medicine is $0.004$ . The relative risk for people aged 70 or over, compared with people under 70, is $2.5$ . A clinic gives the medicine to 3000 people aged 70 or over. Estimate the number who have a reaction.
[3 marks]
Answer

Exam-style questions

D1
Sian rolls a dice 12 times and does not get a six. Sian says, "The dice must be biased."
Is Sian correct? Give a reason for your answer.
[2 marks]

D2
At a fair, a charity game costs £1 to play. A player picks one of 20 cards at random. One card wins £10, three cards win £2 each and the other cards win nothing.
(a)
Is the game fair to the player? You must show your working.
[3 marks]
(b)
During the day, 150 people play the game. Estimate the profit the charity makes.
[2 marks]
Answer
Total for D2: 5 marks
D3
A study recorded whether people smoke and whether they have a lung condition.
(a)
Work out the relative risk of the lung condition for smokers compared with non-smokers.
[2 marks]
Answer
(b)
Interpret your answer to part (a) in context.
[1 mark]
(c)
A headline says, "Smokers have a 3.5% chance of the lung condition." Explain what is wrong with the headline.
[2 marks]
Total for D3: 5 marks

Extension

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

E1
A game costs £1.50 to play. A player rolls a fair dice. A six wins £ $p$ , a five wins £2, a four wins £1.50 and any other score wins nothing. The game is fair. Work out the value of $p$ .
[3 marks]
Hint 1 · What to try
In a fair game, the average win equals the cost to play.
Hint 2 · The first line
$\frac{1}{6} (p + 2 + 1.5) = 1.5$ .
Hint 3 · The full method
$p + 3.5 = 9$ , so $p = 5.50$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can estimate a probability from relative frequency and say whether something is biased.
I can work out expected frequency, absolute risk and relative risk, and interpret them.
I can decide whether a game is fair.

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