In a game, $2$ red balls and $8$ blue balls are placed in a bottle The bottle is shaken and Mary dra

Question

In a game, $2$ red balls and $8$ blue balls are placed in a bottle The bottle is shaken and Mary draws $3$ balls at random and without replacement. The number of red balls that she draws is denoted by $R$. Find the probability distribution of $R$ and show that $\mathrm{P}\left( R\ge 1 \right)=\frac{8}{15}$. Show that the expectation of $R$ is $\frac{3}{5}$ and find the variance of $R$. Mary scores $4$ points for each red ball that she draws. The balls are now replaced in the bottle and the bottle is shaken again. John draws $3$ balls at random and without replacement. He scores $1$ point for each blue ball that he draws. Mary’s score is denoted by $M$ and John’s score is denoted by $J$. Find the expectation and variance of $M-J$.

Accepted Answer

(i) $P(R=0)=\frac7{15},P(R=1)=\frac7{15},P(R=2)=\frac1{15}$, $P(R\ge1)=\frac8{15}$, $E(R)=\frac35$, $\operatorname{Var}(R)=\frac{28}{75}$; (ii) $\frac{476}{75}$

N1999 P2 Q7

N1999 P2 Q7

Video Solution:

Handwritten Solution: