Paper 2 Section B Practice: Probability

Original practice material

These questions total 22 marks. Give probabilities to at least 3 significant figures where appropriate.

Answers: Probability Answers.

Question 1: conditional probability and independence

Events $A$ and $B$ satisfy

$$P(A)=0.45,P(B)=0.60,P(A\cap B)=\mathrm{0.30.}$$

(a) Find $P(A\mid B)$. [2]

(b) Find $P(A\cup B)$. [2]

(c) Determine whether $A$ and $B$ are independent, giving a probability comparison to justify your answer. [2]

Question 2: sampling without replacement

A bag contains 5 red, 3 blue and 2 green counters. Two counters are drawn at random without replacement.

(a) Find the probability that the counters have the same colour. [3]

(b) Given that at least one counter is red, find the probability that both counters are red. [3]

(c) Let $R_1$ and $R_2$ denote the events that the first and second counters are red. Determine whether $R_1$ and $R_2$ are independent. [2]

Question 3: total probability and reverse conditioning

Components are supplied by factories $X$ and $Y$. Factory $X$ supplies $65\%$ of the components and factory $Y$ supplies the rest. A component from $X$ is defective with probability $0.018$, while a component from $Y$ is defective with probability $0.032$.

(a) Find the probability that a randomly selected component is defective. [3]

A screening device flags a defective component with probability $0.94$ and flags a non-defective component with probability $0.05$.

(b) Find the probability that a randomly selected component is flagged. [3]

(c) Given that a component is flagged, find the probability that it was supplied by factory $Y$. [2]