Two Red Balls

This is a hypergeometric probability problem.

Use $\frac{\left(\genfrac{}{}{0ex}{}{3}{2}\right)\left(\genfrac{}{}{0ex}{}{7}{3}\right)}{\left(\genfrac{}{}{0ex}{}{10}{5}\right)}$.

1. 1. Hypergeometric setup

   Choose 2 of 3 red and 3 of 7 blue, out of choosing 5 from 10 total.

2. 2. Compute counts

   $\left(\genfrac{}{}{0ex}{}{3}{2}\right)=3$, $\left(\genfrac{}{}{0ex}{}{7}{3}\right)=35$, $\left(\genfrac{}{}{0ex}{}{10}{5}\right)=252$.

   $$P=\frac{3\times 35}{252}$$

3. 3. Simplify

   $P=\frac{105}{252}=\frac{5}{12}$.

   $$P=\frac{5}{12}$$

Answer: 5/12