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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
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.
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.
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.
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?
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.
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.
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.
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.
8. | 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.
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.
9. | 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.
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