CAIE A-Level Mathematics 5.3.3 Exclusive and Independent Events

Practise distinguishing mutually exclusive from independent events and justifying independence numerically using intersections, marginal probabilities or conditional probabilities.

Exam points

calculate P(A), P(B) and P(A ∩ B) from the table, tree or equally likely outcomes
declare independence only when P(A ∩ B) equals P(A)P(B), showing the numerical comparison
distinguish disjoint events with zero intersection from independent events with unchanged probability

[Maximum number: 3]

Bag A contains 4 balls numbered 2, 4, 5, 8. Bag B contains 5 balls numbered 1, 3, 6, 8, 8. Bag C contains 7 balls numbered 2,7,8,8,8,8,9. One ball is selected at random from each bag.
- Event X is 'exactly two of the selected balls have the same number'.
- Event Y is 'the ball selected from bag A has number 4'.

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

$\mathrm{P}(X \cap \mathrm{Y})=\mathrm{P}(4,8,8)=\frac{1}{4} \times \frac{2}{5} \times \frac{4}{7}=\frac{2}{35}$
B1
\mathrm{P}(Y)=\frac{1}{4}
\frac{2}{35} \neq \frac{47}{140} \times \frac{1}{4}
M1
Attempt to compare $\mathrm{P}(X \cap \mathrm{Y})$ with $\mathrm{P}(X) \times \mathrm{P}(Y)$ or
using conditional probabilities
Not independent
A1
Correct answer, correct working only

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5.3.3—Exclusive and independent events question 1