Three rotten apples are accidentally mixed with fifteen good apples.

Question

Three rotten apples are accidentally mixed with fifteen good apples. Taking $X$ to be the number of rotten apples in a draw of two apples, the variance of $X$ is

Accepted Answer

$$P(X=0)=\frac{{}^{15}C_2}{{}^{18}C_2},\quad P(X=1)=\frac{{}^{15}C_1{}^3C_1}{{}^{18}C_2},\quad P(X=2)=\frac{{}^3C_2}{{}^{18}C_2}$$

With ${}^{18}C_2=153$: $P(X=0)=\dfrac{105}{153}$, $P(X=1)=\dfrac{45}{153}$, $P(X=2)=\dfrac{3}{153}$.

$$\mu = \frac{45+6}{153}=\frac{51}{153}=\frac13,\qquad \sigma^2 = \frac{45+12}{153}-\frac19 = \frac{57}{153}-\frac{17}{153}=\frac{40}{153}$$

Solutions · 1