Exam-Style Questions.

1.	GCSE Higher

The following table shows the probabilities of winning certain prizes at a stall in Faulton Towers theme park. A prize is always given for every turn.

Prize	Pencil	Calculator	Teddy Bear	Ball	Protractor
Probability	0.65	0.02	0.05	0.26	?

In the last year Simon has had 200 turns receiving 200 prizes.

Work out an estimate for the total number of times he has won either a pencil or a protractor.

Worked Solution

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2.	GCSE Higher

Derren thinks he is a mind reader and can tell what Paige is thinking. An experiment was conducted to test this.

Derren and Paige sit either side of a screen and Paige rolls a six-sided fair dice. Paige then thinks about the number on the dice while Derren tries to predict this number.

(a) In 900 attempts, how many correct predictions would you expect Derren to make if he was just guessing?

The results of the first 15 attempts are shown in the table:

Paige's Number	4	6	3	3	3	2	2	2	3	1	1	5	1	1	2
Derren's Guess	1	2	2	4	2	2	1	4	2	4	2	6	1	2	2
Guess Correct						✓							✓		✓

(b) Estimate the probability of getting a matching pair using the results of the last five guesses.

(c) Estimate the probability of getting a matching pair using all 15 results.

(d) Use this information to comment on Derren’s belief that he knows what Paige is thinking.

Worked Solution

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3.	GCSE Higher

At the Trantown annual Fayre there are a number of different sideshows designed to raise money for cancer research. Roger is looking after two of these, the lucky dip and the raffle.

(a) At the lucky dip it costs £1 to enter and everyone wins one of three different types of prize.

• The probability of winning a teddy bear is \(x\);
• The probability of winning is soft drink is \(2x\);
• The probability of winning chocolate bar is \(7x\).

Calculate the value of \(x\).

(b) Roger is offering a free raffle ticket for every raffle ticket purchased. He tells people that this doubles their chances of winning. Is he correct?

Worked Solution

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4.	GCSE Higher

The table shows the shoe sizes of 35 male teachers.

Shoe size	Number of teachers
7	4
8	6
9	10
10	12
11	3

Eight of the 35 teachers do not wear socks.

Ralph thinks that if you choose at random one of the 35 teachers, the probability that he either does not wear socks or wears the median shoe size is \( \frac{17}{35} \) because:

$$ \frac{8}{35} + \frac{9}{35} = \frac{17}{35} $$

Is Ralph correct? Give a reason for your answer.

Worked Solution

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5.	GCSE Higher

Aran, Betsy and Connor collect data about which way up a drawing pin will land if dropped from a fixed height. Here are their results.

	Pin up	Pin down	Total
Aran	24	33	57
Betsy	22	33	55
Connor	86	102	188

Aran thinks "As a drawing pin can only land with its pin up or with its pin down so the probability of a drawing pin landing pin up is 0.5"

(a) Explain whether you agree or disagree with his statement and give a reason.

(b) Connor's results give the best estimate of the probability of a pin landing pin up. Explain why.

(c) Two pins are dropped. Estimate the probability that both pins land pin down.

Worked Solution

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6.	GCSE Higher

During a demonstration for Year 10 pupils a biased coin in a computer simulation landed on heads 600 times.

The relative frequency of heads was 0.3

Work out the number of times the coin landed on tails during that demonstration.

Worked Solution

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7.	GCSE Higher

Donna is in a class of 25 students, 5 of whom have a pet dog.

There are 2200 students in the school.

(a) Use this information to estimate how many students in the school have pet dogs.

(b) Luka is at a different school.

She is in a class of 28 students, 7 of whom have pet dogs.

Luka and Donna have an idea. They think "In our two classes there are 53 students, 12 of whom have a pet dog. We can use this bigger sample to improve the estimate for Donna's school."

What assumption have they made?

(c) Use this idea to estimate how many students in Donna's school have pet dogs.

Worked Solution

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8.	IB Analysis and Approaches

On a Tuesday at a theme park, a sample of 50 visitors was randomly selected as they were leaving the park. They were asked how many times that day they had been on a ride called The Serpent. This information is summarised in the following frequency table.

Number of times on The Serpent	Frequency
0	6
1	12
2	17
3	10
4	5

It can be assumed that this sample is representative of all visitors to the park for the following day.

(a) For the following day, Wednesday, estimate

(i) the probability that a randomly selected visitor will ride The Serpent;

(ii) the expected number of times a visitor will ride The Serpent.

It is known that 2000 visitors will attend the amusement park on Wednesday. The Serpent can carry a maximum of 18 people each time it runs.

(b) Estimate the minimum number of times The Serpent must run to satisfy demand.

Worked Solution

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9.	IB Standard

Piers wins one of the many prizes on offer in the school raffle but he does not yet know exactly what the prize is.

• The probability that a prize is something to eat is 0.3
• The probability that a prize is worth more than $10 is 0.2
• The probability that a prize is something to eat or worth more than $10 is 0.4

Find the probability that the prize Piers wins is not both something to eat and worth more than $10 as Piers does not like expensive food.

Worked Solution

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10.	IB Standard

A game at a fayre consists of a players throwing one dart at the board pictured below.

The probability of hitting each region and the points scored for hitting that region is given in this table.

Region	Probability	Points
A	\(\frac{1}{25}\)	50
B	\(\frac{2}{25}\)	\(x\)
C	\(\frac{4}{25}\)	20
D	\(\frac{5}{25}\)	10

(a) Find the probability that the dart does not hit the board.

The player scores points as shown in the table above but they lose 20 points if they miss the board completely.

(b) Given that the game is a fair game, find the value of \(x\), the number of points awarded for hitting the B region of the board.

Worked Solution

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