Question

A box contains five black and five yellow balls. Two balls are withdrawn
randomly.If they are of the same colour, then you win $1.10; if they are of
different colours, then you lose $1.00.Calculate
i. the expected value of the amount you win
ii. the variance of the amount you win.

Answer 1

$P(same\; colors)=\frac{5}{10}*\frac{4}{9}*2=\frac{4}{9}.$

$P(different\;colors)=\frac{5}{10}*\frac{5}{9}*2=\frac{5}{9}.$

i. $E(X)=\frac{4}{9}*1.10-\frac{5}{9}*1.00=-\frac{1}{15}\approx0.0667.$

ii. $\sigma^2=\frac{4}{9}*1.10^2+\frac{5}{9}*1.00^2-(-\frac{1}{15})^2=\frac{82}{75}-\frac{1}{225}=\frac{49}{45}\approx1.0889.$