Two balls are selected at random one by one without replacement...

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 :

Options

A $4$
B $14$
C $13$
D $11$

Correct answer

B. $14$

Step-by-step solution

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}

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