Box $A$ contains five cards numbered $1$, $2$, $2$, $3$ and $3$. Box $B$ contains thre

Question

Box $A$ contains five cards numbered $1$, $2$, $2$, $3$ and $3$. Box $B$ contains three cards numbered $4$, $5$ and $5$. Cards that are numbered $2$, $3$ and $5$ are red; cards that are numbered $1$ and $4$ are blue. A card is drawn from each of the two boxes. If both cards are of the same colour, then the score will be the sum of the numbers on the two cards. If both cards are of different colours, then the score will be the product of the numbers on the two cards. Let $X$ be the score obtained. Show that $\mathrm{P}\left( X=8 \right)=\frac{2}{5}$.[2] Find the probability distribution of $X$.[3] Find $\mathrm{E}\left( X \right)$ and $\mathrm{Var}\left( X \right)$.[3] Find the probability that the mean score of $50$ independent observations of $X$ lies between $7.5$ and $8.5$.[2]

Accepted Answer

(iii) ${\mathrm{Var}(X)=\frac{184}{45}}$ (iv) ${\mathrm{P}\left(7.5<\overline{X}<8.5\right)\approx 0.718}$ ($3$ s.f.)

2024 JPJC P2 Q9

2024 JPJC P2 Q9

Video Solution:

Solution:

Solution locked