# Probability that $5$ balls are black after $3$ black balls are added to urn with four balls, each having an equal probability of being white or black

An urn contains $4$ balls, each having equal probability of being white or black. $3$ black balls are added to the urn. The probability that $5$ balls in the urn are black?

My approach. The probability that the urn now has $5$ black balls is equal to the Probability of the bag having $2$ black balls previously. And since the the probability of a black ball is equal to white ball, so out of $4$ balls 2 will be white and 2 will be Black.So the answer should be $\dfrac{1}{2}$.

Is it correct?

• The distribution in the original urn is probabilistic, the probability that exactly $2$ were black is $\binom 42 \times \left(\frac 12\right)^{4}=\frac 6{16}=\frac 38$. Also, you should specify whether you mean "at least $5$" or "exactly $5$". – lulu Feb 14 '18 at 20:24
• okay thanx, yes i should have specified, it was 'exactly'. – Upstart Feb 14 '18 at 20:26

Since we know that 3 balls are black, we only need to find the probability that among the rest 4 balls exactly 2 of them are black. The probability is $\frac{{n}\choose{2}}{2^4} = \frac{6}{16}=\frac{3}{8}$