Moderate -0.3 This is a straightforward application of the independence definition P(X∩Y) = P(X)P(Y) with simple probability calculations from equally likely outcomes. The sample space is small (36 outcomes), making enumeration manageable, and the question explicitly directs students to test independence rather than requiring them to recognize this approach themselves.
3 A fair six-sided die is thrown twice and the scores are noted. Event \(X\) is defined as 'The total of the two scores is 4'. Event \(Y\) is defined as 'The first score is 2 or 5'. Are events \(X\) and \(Y\) independent? Justify your answer.
Independent method to find \(P(X \cap Y)\) without multiplication, either stated or by listing/circling numbers on a probability space diagram. OR conditional prob with a single fraction numerator
\(P(X) \times P(Y) = P(X \cap Y)\), independent
A1
Numerical comparison and conclusion, www
## Question 3:
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(X) = \frac{3}{36}$ $\left(\frac{1}{12} \text{ oe}\right)$ | B1 | |
| $P(Y) = \frac{12}{36}$ $\left(\frac{1}{3} \text{ oe}\right)$ | B1 | |
| $P(X \cap Y) = \frac{1}{36}$ | M1 | Independent method to find $P(X \cap Y)$ without multiplication, either stated or by listing/circling numbers on a probability space diagram. OR conditional prob with a single fraction numerator |
| $P(X) \times P(Y) = P(X \cap Y)$, independent | A1 | Numerical comparison and conclusion, www |
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3 A fair six-sided die is thrown twice and the scores are noted. Event $X$ is defined as 'The total of the two scores is 4'. Event $Y$ is defined as 'The first score is 2 or 5'. Are events $X$ and $Y$ independent? Justify your answer.\\
\hfill \mbox{\textit{CAIE S1 2019 Q3 [4]}}