With reference to the Venn diagram below, $S$ is the universal set and $A$ and $B$ are non-emp

Question

With reference to the Venn diagram below, $S$ is the universal set and $A$ and $B$ are non-empty proper subsets of $S$. Explain whether $A$ and $B$ are independent, and write down the value of $\mathrm{P}(A|B)$.[2]

A fair die has three sides numbered $2$, two sides numbered $1$ and one side numbered $0$. A game is played by throwing this fair die three times and the score, $X$, is the sum of the numbers obtained.
1. Show that $\mathrm{P}(X=2) = \frac{7}{72}$.[2]
2. The table shows an incomplete probability distribution of $X$. Find the missing probabilities, showing your workings clearly.[3]
  $x$ $0$ $1$ $2$ $3$ $4$ $5$ $6$
  $\mathrm{P}(X = x)$ $\frac{1}{216}$ $\frac{1}{36}$ $\frac{7}{72}$ $\frac{7}{24}$
The game is played $5$ times.
1. Find the probability that exactly one of the games obtained a score of $0$ and exactly $2$ games obtained a score of more than $2$.
  [3]

Accepted Answer

(a) Dependent; $\mathrm P(A\mid B)=1$. (b)(i) $\frac7{72}$ (shown). (ii) $\mathrm P(X=3)=\frac{11}{54},\mathrm P(X=5)=\frac14,\mathrm P(X=6)=\frac18$. (iii) $\frac{11045}{6718464}\approx0.00164$.

2025 SAJC P2 Q7

2025 SAJC P2 Q7

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