CAIE A-Level Mathematics AS 5.3.3 Exclusive and Independent Events Questions

Practise testing whether events are independent using probability relationships and data.

Exam points

test independence using products of probabilities
use frequency tables and event data to justify independence decisions
calculate intersections when events are independent

[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.

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

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CAIE A-Level Mathematics AS 5.3.3 Exclusive and Independent Events Questions question 1