Discussion
An urn contains four balls, each ball having equal probability of being white or black. Three black balls are added to the urn. The probability that five balls in the urn are black is
Answer: B
The probability is 3/8.
After three black balls are added there are seven balls, so five are black exactly when two of the original four are black. Each original ball is black with probability 1/2, independently, so the number of black ones among them is binomial:
\[ \begin{aligned} P &= \binom{4}{2}\left(\frac{1}{2}\right)^{4} \\ &= \frac{6}{16} = \frac{3}{8} \end{aligned} \]
5/7 and 2/7 come from treating 'five of seven black' as a ratio of balls rather than a probability over the unknown colours. 1/2 is the chance for a single original ball, not for exactly two of the four.
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