A-level · Probability

Conditional probabilityExam-style questions

The probability of A given B, P(A|B) = P(A ∩ B) / P(B), as restricting the sample space to B — reading it off a Venn diagram, the multiplication formula P(A ∩ B) = P(B)P(A|B), and testing independence with P(A|B) = P(A).

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28 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
Of $100$ people, $40$ own a car ( $A$ ), $30$ own a bike ( $B$ ) and $10$ own both. A person is chosen at random.
(a) Find $P(A \mid B)$ .
(b) Find $P(B \mid A)$ .
(c) Determine whether owning a car and owning a bike are independent.
Level 5 · 4 marksTry it
2
Box A contains $4$ $4$ red counters and $2$ $2$ blue counters. Box B contains $3$ $3$ red counters and $3$ $3$ blue counters. Box C contains $1$ $1$ red counter and $4$ $4$ blue counters.

A fair six-sided die is thrown. If the die shows $1$ $1$ or $2$ $2$ , box A is chosen. If the die shows $3$ $3$ , $4$ $4$ or $5$ $5$ , box B is chosen. If the die shows $6$ $6$ , box C is chosen. Two counters are then taken at random, without replacement, from the chosen box.
- Event $X$ is 'the two counters taken are the same colour'.
- Event $Y$ is 'box B is chosen'.
(a) Find $\mathrm{P}(X)$ $P (X)$ .

(5 marks)
(b) Find $\mathrm{P}(X \cap Y)$ $P (X \cap Y)$ and hence determine whether or not the events $X$ $X$ and $Y$ $Y$ are independent.

(3 marks)
(c) Find the probability that box A was chosen, given that the two counters taken are the same colour.

(2 marks)
Level 5 · 10 marksTry it