If you roll a fair six-sided die twice, what is the probability of getting a 4 on the first roll and an odd number on the second roll?

Sure, let's break down the problem step by step.

1. Understanding the Problem:

   • We are rolling a fair six-sided die twice.
   • We want to find the probability of getting a 4 on the first roll.
   • We also want to find the probability of getting an odd number on the second roll.

2. Probability of Rolling a 4 on the First Roll:

   • A fair six-sided die has six faces, numbered 1 through 6.
   • The probability of rolling any specific number (including 4) is $\frac{1}{6}$.

3. Probability of Rolling an Odd Number on the Second Roll:

   • The odd numbers on a six-sided die are 1, 3, and 5.
   • There are three odd numbers out of six possible outcomes.
   • The probability of rolling an odd number is $\frac{3}{6} = \frac{1}{2}$.

4. Combining the Probabilities:

   • The events of rolling a 4 on the first roll and rolling an odd number on the second roll are independent events.
   • To find the combined probability of two independent events both occurring, we multiply their individual probabilities.

So, the combined probability is: $\text{Probability of 4 on the first roll} \times \text{Probability of an odd number on the second roll}$ $= \frac{1}{6} \times \frac{1}{2}$ $= \frac{1}{12}$

Therefore, the probability of getting a 4 on the first roll and an odd number on the second roll is $\frac{1}{12}$.