6. Suppose that Ethan has four red chips and two white chips. He selects three chips at random and places them in Urn 1, while the remaining chips are placed in Urn 2. He then lets his brother Josh draw one chip from each urn at random. What is the probabiity that the chips drawn by Josh are both red?
Answer: $\frac{2}{5}$
$\underline{\text { Solution: The possibilities are }}$
\begin{tabular}{ccc}
\hline Urn 1 & Urn 2 & Number of Ways \\
\hline 1R, 2W & $3 \mathrm{R}$ & $\binom{4}{1}\binom{2}{2}=4$ \\
$2 \mathrm{R}, 1 \mathrm{~W}$ & $2 \mathrm{R}, 1 \mathrm{~W}$ & $\binom{4}{2}\binom{2}{1}=12$ \\
$3 \mathrm{R}$ & $1 \mathrm{R}, 2 \mathrm{~W}$ & $\binom{4}{3}\binom{2}{0}=4$ \\
\hline
\end{tabular}

The total number of ways to split the chips into the two urns is twenty. Hence, the probability of getting both red chips is
$$
\frac{4}{20}\left(\frac{1}{3}\right)+\frac{12}{20}\left(\frac{2}{3}\right)\left(\frac{2}{3}\right)+\frac{4}{20}\left(\frac{1}{3}\right)=\frac{2}{5} .
$$