# Exam-Style Questions on Probability

## Problems on Probability adapted from questions set in previous exams.

### 1.

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.

### 2.

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?

### 3.

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:

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

### 4.

GCSE Higher

(a) There are 24 people at a Football Club supporters' meeting in Dudley. Thirteen of them are female.

First chosen                         Second chosen

Tracey draws the tree diagram above to show how two people could be chosen from the meeting at random. What is wrong with the probabilities shown on the diagram?

(b) Aynuk and Ayli play for Dudley Casuals Football Club. The probabilities that they will score a goal in the next match are 0.2 and 0.35 respectively.

The manager thinks that the probability that both players will score a goal in the next match is 0.2 + 0.35. Is the manager correct? Give reasons 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.

IGCSE Extended

The letters of the word SUMMER are printed on 6 cards.

(a) A card is chosen at random. Write down the probability that it has the letter R printed on it.

(b) Find the probability that the card chosen at random does not have the letter M printed on it.

(c) Find the probability that the letter printed on the card chosen at random has no lines of symmetry.

(d) Chris chooses a card at random, replaces it, then chooses a card again. Calculate the probability that only one of the cards she chooses has the letter M printed on it.

(e) Graham chooses a card at random and does not replace it. He continues until he chooses a card with the letter M printed on it. Find the probability that this happens when he chooses the 3rd card.

### 7.

GCSE Higher

A box contains only green and yellow crayons.

Luka takes one crayon at random from the box. The probability that she takes a green crayon is $$\frac16$$.

Luka returns the crayon to the bag and adds four more green crayons to the box. The probability that she takes one green crayon at random is now $$\frac14$$.

How many crayons of each colour were originally in the box?

### 8.

GCSE Higher

Lisa's nephew Sean has two boxes containing plastic building bricks.

A red box contains 5 green bricks and 10 yellow bricks.

A blue box contains 8 green bricks and 16 yellow bricks.

Lisa picks a brick at random from the red box and puts it into the blue box.

Sean now picks a brick at random from the blue box. Show that:

P(Lisa picks a green brick) = P(Sean picks a green brick)

### 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.

### 10.

IB Standard

Jane and David play two games of golf. The probability that Jane wins the first game is $$\frac56$$. If Jane wins the first game, the probability that she wins the second game is $$\frac67$$.

If Jane loses the first game, the probability that she wins the second game is $$\frac34$$

(a) Copy and complete the following tree diagram.

(b) Find the probability that Jane wins the first game and David wins the second game.

(c) Find the probability that David wins at least one game.

(d) Given that David wins at least one game, find the probability that he wins both games.

### 11.

IB Standard

The weights in grams of 98 mice are shown in the cumulative frequency diagram. The heaviest mouse weighted 160g.

(a) Write down the median weight of the mice.

(b) Find the percentage of mice that weigh 70 grams or less.

The same data is presented in the following table.

 Weights w grams Freequency 0 < w ≤ 40 40 < w ≤ 80 80 < w ≤ 120 120 < w ≤ 160 p 63 q 3

(c) Find the value of p.

(d) Find the value of q.

(e) Use the values from the table to estimate the mean and standard deviation of the weights.

A second batch of mice are normally distributed with the same mean and standard deviation as those of the first group mentioned above.

(f) Find the percentage of the second batch of mice that weigh 70 grams or less.

(g) A sample of five mice is chosen at random from the second batch. Find the probability that at most three mice weigh 70 grams or less.

### 12.

IB Studies

Neal is attending a Scout jamboree in Japan. He has both boots and trainers to wear. He also has the choice of wearing a cap or not.

The probability Neal wears boots is 0.4. If he wears boots, the probability that he wears a cap is 0.7.

If Neal wears trainers, the probability that he wears a cap is 0.25.

The following tree diagram shows the probabilities for Neal's clothing options at the jamboree.

(a) Find the value of A.

(b) Find the value of B.

(c) Find the value of C.

(d) Calculate the probability that Neal wears trainers and no cap.

(e) Calculate the probability that Neal wears no cap.

(f) Calculate the probability that Neal wears trainers given that he is not wearing a cap.

(g) Calculate the probability that Neal wears boots on the first two days of the jamboree.

(h) Calculate the probability that Neal wears boots on one of the first two days, and trainers on the other.

### 13.

GCSE Higher

Percy throws a fair dice until he gets a five.

(a) Work out the probability that Percy gets the five on his first throw.

(b) Find the probability that he has to throw the dice exactly twice to get the five.

(c) Calculate the probability that he will have the throw the dice more than twice to get a five.

### 14.

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."

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

### 15.

IB Standard

At the south end of Playfair Amusement park are two games of chance, Beat The Goalie and Basketball Bounce.

For the Goalie game, the probability of winning is 0.1. If someone plays this game nine times:

(a) Find the probability that they win exactly three times.

For the Basketball game, the probability of winning is $$x$$. If someone plays it nine times.

(b) Write down an expression, in terms of $$x$$, for the probability of winning exactly three games.

(c) Hence, find the values of $$x$$ such that the probability of winning exactly three games is 0.2.

### 16.

IB Studies

Julie chooses a cake from a yellow box on a shelf. The box contains two chocolate cakes and three plain cakes. She eats the cake and chooses another one from the box. The tree diagram below represents the situation with the four possible outcomes where C stands for chocolate cake and P for plain cake.

(a) Write down the value of $$x$$.

(b) Write down the value of $$y$$.

(c) Find the probability that both cakes are plain.

On another shelf there are two boxes, one red and one green. The red box contains four chocolate cakes and five plain cakes and the green box contains three chocolate cakes and four plain cakes. Ben randomly chooses either the red or the green box and randomly selects a cake.

(d) Copy and complete the tree diagram below.

(e) Find the probability that he chooses a chocolate cake.

(f) Find the probability that he chooses a cake from the red box given that it is a chocolate cake.

### 17.

IB Standard

Let $$J$$ and $$T$$ be independent events, where $$P(J)=0.4$$ and $$P(T)=0.7$$.

(a) Find $$P(J\cap T)$$.

(b) Find $$P(J\cup T)$$.

(c) Draw a two-set Venn diagram and shade the region that represents $$J\cap T'$$.

(d) Find $$P(J\cap T')$$.

### 18.

IB Standard

Eels are elongated fish, ranging in length from 5cm to 4m. In a certain lake the length of the eels are normally distributed with a mean of 84cm and a standard deviation of 18cm. Eels are classified as giant eels if they are more than 120 cm long.

An eel is selected at random from the lake.

(a) Find the probability that this eel is a giant.

(b) Given that this eel is a giant, find the probability that it is longer than 130cm.

(c) Two eels are selected at random. Find the probability that they are both giants.

100 eels are selected at random.

(d) Find the expected number of these eels that are giants.

(e) Find the probability that at least 5 of these eels are giants.

### 19.

GCSE Higher

On the day after one GCSE paper Twitter and the media were buzzing with comments about a particular question about Hanna's sweets. Here is our version:

Hannah has 6 orange sweets and some yellow sweets.

Overall, she has $$n$$ sweets.

The probability of her taking 2 orange sweets is $$\frac13$$.

Prove that: $$n^2-n-90=0$$

Hence or otherwise find $$n$$.

### 20.

GCSE Higher

When the arrow is spun once on this spinner, a 1 or a 2 or a 3 can be scored depending on the colour the arrow lands on.

Aimee is going to spin the arrow twice. She will work out her total score by adding the two scores she gets on the two spins. The probability that she will get a total score of 4 is $$\frac{7}{25}$$

Find the value of $$x$$.

### 21.

GCSE Higher

A bag contains balls that are red, blue, green or yellow.

• The number of red balls is $$x-3$$.
• The number of blue balls is $$2x$$.
• The number of green balls is $$7$$.
• The number of yellow balls is $$5x$$.

A counter is chosen at random. The probability it is green is $$\frac{7}{52}$$

Work out the probability it is red.

### 22.

GCSE Higher

There are $$x$$ left shoes and 7 right shoes in a dark cupboard.

Harper takes at random two shoes from the cupboard.

The probability that Harper takes one left shoe and one right shoe is $$\frac{7}{13}$$

(a) Show that $$x^2-13x+42= 0$$

(b) Find the probability that Harper takes two right shoes.

The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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Kausar Begum Khan, Kenya

Friday, January 26, 2018

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