A-level · Probability

Tree diagrams and sequential eventsExam-style questions

Using a tree diagram for two-stage experiments: multiply probabilities ALONG the branches of a path (AND), add across the paths that make up an event (OR), and use 1 − P(none) for "at least one". The crucial distinction between sampling WITH replacement (probabilities stay the same) and WITHOUT replacement (the totals, and so the later probabilities, change).

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12 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Tin $A$ contains 5 plain biscuits and 3 chocolate biscuits only.

Tin $B$ contains 2 plain biscuits and 4 chocolate biscuits only.

Two biscuits are taken at random, one at a time without replacement, from tin $A$ and are placed in tin $B$ . A biscuit is then taken at random from tin $B$ .

The tree diagram below, where $r$ , $s$ , $t$ and $u$ are probabilities, shows this information. $P$ represents a plain biscuit and $C$ represents a chocolate biscuit.(a) Complete the tree diagram by finding the values of $r$ , $s$ , $t$ and $u$

(3 marks)
(b) Find the probability that the two biscuits placed in tin $B$ are of different types.

(2 marks)
(c) Find the probability that all three biscuits taken are of the same type.

(2 marks)

Given that the biscuit taken from tin $B$ is plain,
(d) find the probability that both of the biscuits placed in tin $B$ were plain.

(3 marks)
Level 4 · 10 marksTry it
2
Elena is an archer. In each round of a practice session she shoots arrows at a target, one at a time.
- The probability that the first arrow of a round hits the centre of the target is $0.4$
- After an arrow that hits the centre, the probability that her next arrow hits the centre is $0.65$
- After an arrow that misses the centre, the probability that her next arrow hits the centre is $0.25$
A round ends as soon as two of Elena's arrows have hit the centre, or when she has shot 3 arrows.

In the tree diagram below, which shows the possible outcomes of a round, $r$ $r$ , $s$ $s$ , $t$ $t$ and $u$ $u$ are probabilities. $H$ $H$ represents an arrow that hits the centre and $M$ $M$ represents an arrow that misses the centre.(a) Complete the tree diagram by finding the values of $r$ $r$ , $s$ $s$ , $t$ $t$ and $u$ $u$

(3 marks)
(b) Find the probability that a round ends after Elena's 2nd arrow.

(2 marks)
(c) Find the probability that exactly 2 of Elena's arrows in a round hit the centre.

(2 marks)

Given that exactly 2 of Elena's arrows in a round hit the centre,
(d) find the probability that her first arrow in that round missed the centre.

(2 marks)

Given that Elena shoots 3 arrows in a round,
(e) find the probability that exactly 1 of these arrows hits the centre.

(2 marks)
Level 4 · 11 marksTry it
3
A jar contains only lemon sweets and lime sweets. There are $n$ lemon sweets in the jar, and there are 4 more lime sweets than lemon sweets.

Jo takes a sweet at random from the jar and eats it. She then takes a second sweet at random from the jar and eats it.
(a) Complete the tree diagram below, giving each probability in terms of $n$ (3 marks)
(b) Show that the probability that at least one of the two sweets Jo takes is lemon is

$\frac{n(3n + 7)}{2(n + 2)(2n + 3)}$

(2 marks)

The probability that at least one of the two sweets Jo takes is lemon is $\frac{5}{8}$
(c) Find the number of sweets in the jar before Jo took any.

(2 marks)
(d) Given that at least one of the two sweets Jo takes is lime, find the probability that the first sweet she takes is lemon. You must show your working.

(2 marks)
Level 4 · 9 marksTry it
4
Megan and Ewan play a game in which an ordinary fair dice is thrown four times.
- Megan scores one point for every throw that shows $3$ , $4$ , $5$ or $6$ .
- Ewan scores one point for every throw that shows $1$ or $2$ .
- The game is won by the player with at least three points. If the points are shared two each, the game is drawn.
(a) Determine

(i) the probability that Megan wins the game,

(ii) the probability that the game is drawn.

(5 marks)

After a drawn game, Megan and Ewan play again, and they carry on playing games until one of them wins a game.
(b) Determine the probability that Ewan is the first to win a game.

(3 marks)
Level 4 · 8 marksTry it