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Miscellaneous Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. The questions shown here or their solutions contain the text 'Tree Diagram'. |
1. | GCSE Higher |
The midnight train to Georgia is not the most reliable. The probability that the train will be late on any day is 0.35
(a) Complete the probability tree diagram for Monday and Tuesday.
Monday Tuesday
(b) Work out the probability that the train will be late on at least one of these two days.
2. | GCSE Higher |
Carol has two fair 5-sided spinners. One spinner is pink and the other is green.
The spinners could land on any of the single digit odd numbers with equal probability.
(a) Draw a tree diagram to show the probabilities of either, or both spinners landing on a square number.
(b) Work out the probability that the pink spinner lands on a square number and the green spinner does not.
3. | GCSE Higher |
Near Blizzard Lodge on any day the probability that it snows is \( \frac{1}{4} \).
When it snows the probability that Angus goes skiing is \( \frac{4}{7} \).
When it does not snow the probability that Angus goes skiing is \( \frac{2}{5} \).
(a) In a period of 80 days on how many days is it expected to snow?
(b) Complete the tree diagram.
(c) Find the probability that on any day Angus goes skiing.
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. | 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.
6. | 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.
7. | IB Applications and Interpretation |
The Scrumptious Sweet Company sell a variety pack of colourful, shaped sweets. The sweets are produced such that 60% are square and 40% are circular. It is known that 20% of the square shaped sweets and 40% of the circular sweets are coloured red.
(a) Show this information in a tree diagram.
A sweet is selected at random.
(b) Find the probability that the sweet is red.
(c) Given that the sweet is red, find the probability it is circular.
The Scrumptious Sweet Company also produce variety packs of Rainbow Gums. Their specifications state that the colours in each variety pack should be distributed as follows.
| Colour | Red | Orange | Yellow | Green | Blue |
| Percentage (%) | 15 | 25 | 15 | 25 | 20 |
Inspector Lou Spowels opens a pack of 90 sweets and records the frequency of each colour:
| Colour | Red | Orange | Yellow | Green | Blue |
| Observed Frequency | 12 | 21 | 15 | 20 | 22 |
To investigate if the sample is consistent with the company's specifications, Mr Spowels conducts a \(\chi^2\) goodness of fit test. The test is carried out at a 5% significance level.
(d) Write down the null hypothesis for this test.
(e) Copy and complete the following table giving the frequencies correct to one decimal place:
| Colour | Red | Orange | Yellow | Green | Blue |
| Expected Frequency |
(f) Write down the number of degrees of freedom.
(g) Find the p-value for the test.
(h) State the conclusion of the test. Give a reason for your answer.
8. | 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.
9. | 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.
10. | A-Level |
Mathsland's national currency comes in denominations of 1 unit, 5 units, 10 units and 50 units. Sofya places this collection of these coins in her purse then, without looking, takes out two coins at random, one after the other.
Draw a tree diagram to represent the situation and then use it to calculate the probability that the second coin that Sofya takes out of her purse has a greater value than the first.
11. | IB Analysis and Approaches |
Some orange fish and green fish are swimming in a large tank in the enterance to a restaurant. The number of orange fish is a single digit number as is the number of green fish.
The chef occasionally takes a fish from the tank at random. So far today he has taken out two fish.
The tank initially contains r orange fish and g green fish.
Let \( P(GG) \) represent the probability of drawing two green fish from the tank without replacement.
It is known that \( P(GG) = \frac{1}{5} \).
(a) Show that \( 4g^2 - (4 + 2r)g + r - r^2 = 0 \).
(b) By solving the equation in part (a), show that \( g = \dfrac{(2+r) \pm \sqrt{5r^2 + 4}}{4} \).
(c) Find two pairs of values for r and g that satisfy the condition \( P(GG) = \frac{1}{5} \).
On a different day the chef randomly takes three fish out of the tank. The tank initially contained 10 orange fish and g green fish.
Let \( P(GGG) \) represent the probability of taking three green fish from the tank without replacement.
(d) Find an expression for \( P(GGG) \) in terms of g.
A green fish is added so that the tank now contains 10 orange fish and \( g + 1 \) green fish. The probability of taking three green fish from the tank without replacement is now twice the probability expressed in part (d).
(e) Find the initial number of green fish in the tank on this particular day.
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