A bag contains 6 balls.

Question

A bag contains 6 balls. Two balls are drawn at random and both found to be black. The probability that the bag contains at least 5 black balls is

Accepted Answer

Let $E_{i} \rightarrow$ Bag have at least $i$ black balls

$E \rightarrow 2$ balls are drawn \& both black

$$
\begin{aligned}
& \therefore P\left(\frac{E_{5} \text { or } E_{6}}{E}\right)=\frac{P\left(\frac{E}{E_{5}}\right)+P\left(\frac{E}{E_{6}}\right)}{\sum\limits_{i=1}^{6} P\left(\frac{E}{E_{i}}\right)} \\
& =\frac{\frac{{ }^{5} C_{2}}{{ }^{6} C_{2}}+\frac{{ }^{6} C_{2}}{{ }^{6} C_{2}}}{0+\frac{{ }^{2} C_{2}}{{ }^{6} C_{2}}+\frac{{ }^{3} C_{2}}{{ }^{6} C_{2}}+\frac{{ }^{4} C_{2}}{{ }^{6} C_{2}}+\frac{{ }^{5} C_{2}}{{ }^{6} C_{2}}+\frac{{ }^{6} C_{2}}{{ }^{6} C_{2}}} \\
& =\frac{10+15}{1+3+6+10+15}=\frac{25}{35}=\frac{5}{7}
\end{aligned}
$$

Solutions · 1