Three rotten apples are accidently mixed with fifteen good apples....

Given apples are: $3$ bad, $15$ good

Let $X$ be no. of bad apples.

$⇒ P\left(X = 0\right) = \dfrac{{}^{15}{C}_{2}}{{}^{18}{C}_{2}} = \dfrac{\dfrac{15 !}{2 ! \times 13 !}}{\dfrac{18 !}{2 ! \times 16 !}}$

$⇒ P\left(X = 0\right) = \dfrac{15 \times 14}{18 \times 17} = \dfrac{5 \times 7}{3 \times 17}$

$⇒ P\left(X = 0\right) = \dfrac{35}{51}$

$⇒ P\left(X = 1\right) = \dfrac{{}^{3}{C}_{1}{\times}^{15}{C}_{1}}{{}^{18}{C}_{2}} = \dfrac{3 \times 15}{9 \times 17}$

$⇒ P\left(X = 1\right) = \dfrac{5}{17}$

$⇒ P\left(X = 2\right) = \dfrac{{}^{3}{C}_{2}}{{}^{18}{C}_{2}} = \dfrac{3}{9 \times 17}$

$⇒ P\left(X = 2\right) = \dfrac{3}{153}$

$⇒ E\left(X\right) = 0 \times \dfrac{35}{51} + 1 \times \dfrac{5}{17} + 2 \times \dfrac{3}{153}$

$⇒ E\left(X\right) = \dfrac{1}{3}$

We know that, $Var\left(X\right) = E\left({X}^{2}\right) - {\left(E\left(X\right)\right)}^{2}$

$⇒ Var\left(X\right) = 0 \times \dfrac{35}{51} + 1 \times \dfrac{5}{17} + 4 \times \dfrac{1}{51} - {\left(\dfrac{1}{3}\right)}^{2}$

$⇒ Var\left(X\right) = \dfrac{19}{51} - \dfrac{1}{9}$

$⇒ Var\left(X\right) = \dfrac{40}{153}$