Question

Four bad oranges are accidently mixed with 16 good ones. Find the probability distribution of the number of bad oranges when two oranges are drawn at random from this lot. Find the mean and variance of the distribution.

Answer 1

Let X denotes the random variable. Then X can take values 0, 1, 2.
The probability distribution is as follow :
$\begin{array}{cccc}X& 0& 1& 2\\ P\left(X\right)& \frac{{}^{16}{C}_2}{{}^{20}{C}_2}=\frac{60}{95}& \frac{{}^4{C}_1{}^{16}{C}_1}{{}^{20}{C}_2}=\frac{32}{95}& \frac{{}^4{C}_2}{{}^{20}{C}_2}=\frac{3}{95}\end{array}$
Now mean, $\mu =\sum XP\left(X\right)=0+\frac{32}{95}+\frac{6}{95}=\frac{38}{95}=\frac{2}{5}.$
Variance, ${\delta }^2=\sum X^{2}P\left(X\right)-{\mu }^2=0+\frac{32}{95}+\frac{12}{95}=\frac{44}{95}-\frac{4}{25}=\frac{144}{475}.$