Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$, then $m+n$ is equal to :

Question

Aicharya · Accepted Answer

Bag contains 4 white and 6 black balls
$A$ : first ball selected is black
$B$ : Second ball is also black
$\begin{aligned}
& P\left(\frac{A}{B}\right)=\frac{\frac{6}{10} \times \frac{5}{9}}{\frac{4}{10} \times \frac{6}{9}+\frac{6}{10} \times \frac{5}{9}}=\frac{30}{24+30} \\ & =\frac{30}{54}=\frac{5}{9} \\ & m+n=5+9=14
\end{aligned}$ /

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black…

Solution