Step 1:

Total number of oranges=12

Number of bad oranges=2

Let $X$ be the discrete RV denoting the number of bad oranges when 3 oranges are drawn from the lot of 12

$\Rightarrow\:X$ takes the values $0,1,2$

Step 2:

$P(X=0)$=probability of no bad oranges

$\qquad\quad\;\;=\large\frac{10C_3}{12C_3}$

$\qquad\quad\;\;=\large\frac{10\times 9\times 8/1\times 2\times 3}{12\times 11\times 10/1\times 2\times 3}$

$\qquad\quad\;\;=\large\frac{36}{66}$

$\qquad\quad\;\;=\large\frac{6}{11}$

$P(X=1)$=probability of drawing 1 bad orange

$\qquad\quad\;\;=\large\frac{2\times 10}{12\times 11/1\times 2}$

$\qquad\quad\;\;=\large\frac{20}{66}$

$\qquad\quad\;\;=\large\frac{10}{33}$

$P(X=2)$=probability of drawing 2 bad oranges

$\qquad\quad\;\;=\large\frac{2C_2\times 10C_1}{12C_2}$

$\qquad\quad\;\;=\large\frac{1\times 10}{12\times 4/1\times 2}$

$\qquad\quad\;\;=\large\frac{10}{66}=\frac{5}{33}$

Step 3:

The probability distribution is given by

Step 4

Expected value of $X$ is Mean $=\sum_{x=0}^{2}\:x.P(x)$

Mean$=(0\times\large\frac{6}{11})$$+(1\times\large\frac{10}{33})$$+(2\times\large\frac{5}{33})$

$=\large\frac{20}{33}$