What is the probability of rolling a sum of nine on the first throw and a sum of six on the second throw of two fair dice?

When two fair dice are thrown, each die has 6 possible outcomes. The total number of possible outcomes is $6 \times 6 = 36$.

The combinations that result in a sum of nine are:
$(3, 6)$, $(4, 5)$, $(5, 4)$, $(6, 3)$
There are 4 favorable outcomes for a sum of nine.
The probability of getting a sum of nine, $P(\text{sum=9})$, is:

The combinations that result in a sum of six are:
$(1, 5)$, $(2, 4)$, $(3, 3)$, $(4, 2)$, $(5, 1)$
There are 5 favorable outcomes for a sum of six.
The probability of getting a sum of six, $P(\text{sum=6})$, is:

Since the two throws are independent events, the probability of both events happening is the product of their individual probabilities:

The probability of obtaining a sum of nine in the first throw and a sum of six in the second throw is $\boxed{\frac{5}{324}}$.