There are 5 black balls and 5 white balls in a box. A person picks 5 balls from the box at random.
What is the probability that exactly 3 of 5 balls are white?
Hypergeometric distribution:
\(N\) is the population size \(= 10\),
\(K\) is the number of succeess states in the population \(= 5\),
\(n\) is the number of draws(i.e. quantitiy drawn in each trial) \(= 5\),
\(k\) is the number of abserved successes \(= 3\).
\(\begin{array}{|rcll|} \hline && \dfrac{ \binom{K}{k} \binom{ N-K}{n-k} }{ \binom{N}{n} } \\\\ &=& \dfrac{ \binom{5}{3}_{\text{white}} \binom{5}{2}_{\text{black}} } { \binom{10}{5}_{\text{all}} } \quad & | \quad \binom{5}{3}=\binom{5}{5-3}=\binom{5}{2} \\\\ &=& \dfrac{ \binom{5}{2} \binom{5}{2} } { \binom{10}{5} } \quad & | \quad \binom{5}{2}= \frac52\cdot \frac41 = 10 \\\\ &=& \dfrac{ 10\cdot 10 } { \binom{10}{5} } \quad & | \quad \binom{10}{5}=\frac{10}{5}\cdot \frac94 \cdot \frac83\cdot \frac72 \cdot \frac61 = 252 \\\\ &=& \dfrac{ 100 } { 252 } \\\\ &=& \dfrac{ 50 } { 126 } \\\\ &=& \dfrac{ 25 } { 63 } \\\\ &=& 0.39682539683\quad (=39.7\ \%) \\ \hline \end{array}\)
Source see: https://en.wikipedia.org/wiki/Hypergeometric_distribution