Does anyone know how to find the probability of selecting a jury of $12$ people ($6$ men and $6$ women) out of an initial group of $18$ people ($6$ men and $12$ women)?
With my knowledge of probability, the only way I can think to solve this is by drawing a big tree diagram with $12$ events. Is there a quicker way to solve this problem?
There are $\binom{6}{6}=1$ ways to choose $6$ men, and $\binom{12}{6}=924$ ways to choose the $6$ women. Therefore, there are $\binom{6}{6}\cdot\binom{12}{6}=924$ ways to pick $12$ jurors so that $6$ are women and $6$ are men. In total, there are $\binom{18}{12}=18564$ ways to choose $12$ jurors from $18$ people. Hence, the desired probability is
$$ \frac{924}{18564}\approx 4.98\%. $$