Out of seven Easter eggs, three are red. We placed ten eggs into a larger box and six into a smaller box randomly. What is the probability that there is a red egg in both boxes?
Solution. We can choose 6 eggs out of 16 in $\binom{16}{6}$ ways. This is the total number of cases.

First, let's put 1 red and 5 non-red eggs into the smaller box. There are 3 ways to choose the 1 red egg, and we can choose the other 5 non-red eggs in $\binom{13}{5}$ ways. This gives us a total of $3 \cdot\binom{13}{5}$ possibilities.

If we want to put 2 red eggs into the smaller box (to ensure that at least 1 red egg remains in the larger box), the number of favorable cases is: $\binom{3}{2}\binom{13}{4}$. Thus, the desired probability is:

$$
\frac{3 \cdot\binom{13}{5}+\binom{3}{2}\binom{13}{4}}{\binom{16}{6}}=\frac{3 \cdot 1287+3 \cdot 715}{8008}=\frac{6006}{8008}=\frac{3}{4}
$$

The probability that each box contains a red egg is $\frac{3}{4}$.