Question

A single die is rolled twice. Find the probability of rolling an odd number the first time and a number greater than 4 the second time.

Answer 1

To find the probability of rolling an odd number on the first roll of a die and a number greater than 4 on the second roll, we can analyze each event separately since the two rolls are independent.

1. **Rolling an odd number on the first roll:**
The odd numbers on a standard six-sided die are 1, 3, and 5. Thus, there are 3 odd numbers out of the 6 possible outcomes.

The probability of rolling an odd number on the first roll is:
\[
P(\text{odd first roll}) = \frac{3}{6} = \frac{1}{2}
\]

2. **Rolling a number greater than 4 on the second roll:**
The numbers greater than 4 on a die are 5 and 6. Thus, there are 2 numbers greater than 4 out of the 6 possible outcomes.

The probability of rolling a number greater than 4 on the second roll is:
\[
P(\text{greater than 4 second roll}) = \frac{2}{6} = \frac{1}{3}
\]

3. **Calculating the joint probability:**
Since the two rolls are independent, we can multiply the probabilities of the two events:
\[
P(\text{odd first roll and greater than 4 second roll}) = P(\text{odd first roll}) \times P(\text{greater than 4 second roll})
\]
\[
= \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}
\]

Therefore, the probability of rolling an odd number the first time and a number greater than 4 the second time is \(\boxed{\frac{1}{6}}\).