Three defective oranges are accidentally mixed with seven good ones, indistinguishable by …

Question

Three defective oranges are accidentally mixed with seven good ones, indistinguishable by appearance. Two oranges are drawn at random from the lot. If $X$ denotes the number of defective oranges, then the variance of $X$ is

Accepted Answer

$$P(X=0)=\frac{{}^7C_2}{{}^{10}C_2}=\frac{42}{90\cdot\frac12}\ \text{i.e.}\ \frac{21}{45},\quad P(X=1)=\frac{{}^7C_1{}^3C_1}{{}^{10}C_2}=\frac{21}{45},\quad P(X=2)=\frac{{}^3C_2}{{}^{10}C_2}=\frac{3}{45}$$

$$\mu = \sum x_iP_i = \frac{21+6}{45}=\frac{27}{45}=\frac35$$

$$\sigma^2 = \sum x_i^2P_i-\mu^2 = \frac{21+12}{45}-\left(\frac35\right)^2=\frac{66}{90}-\frac{9}{25}=\frac{28}{75}$$

Solutions · 1