If two fair dice are thrown together twice, find the probability of obtaining a sum of nine in the first throw and a sum of six in the second throw.

For a single throw of two dice, the total number of outcomes is $6 \times 6 = 36$.

The pairs that sum to nine are $(3, 6), (4, 5), (5, 4), (6, 3)$. There are 4 favorable outcomes.
The probability of a sum of nine is $P(\text{sum of 9}) = \frac{4}{36} = \frac{1}{9}$.

The pairs that sum to six are $(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)$. There are 5 favorable outcomes.
The probability of a sum of six is $P(\text{sum of 6}) = \frac{5}{36}$.

The probability is $\frac{5}{324}$.