Let's address the given problem step by step.
1. Understanding the Events:
- We are given a six-sided die, which has the numbers: 1, 2, 3, 4, 5, and 6.
- We need to find the probability of rolling a prime number, given that the number rolled is even.
2. Identify Even and Prime Numbers on a Six-Sided Die:
- Even numbers on a six-sided die: 2, 4, 6.
- Prime numbers on a six-sided die: 2, 3, 5.
- Notice that the only number that is both even and prime is 2.
3. Defining the Probabilities:
- Let [tex]\( P(P \cap E) \)[/tex] represent the probability of rolling a number that is both prime and even.
- Let [tex]\( P(E) \)[/tex] represent the probability of rolling an even number.
4. Compute [tex]\( P(P \cap E) \)[/tex]:
- There is only one outcome (2) that is both prime and even out of six possible outcomes on the die.
- Thus, [tex]\( P(P \cap E) = \frac{1}{6} \)[/tex].
5. Compute [tex]\( P(E) \)[/tex]:
- There are three even numbers (2, 4, 6) out of the six possible outcomes on the die.
- Thus, [tex]\( P(E) = \frac{1}{2} \)[/tex].
6. Applying Conditional Probability Formula:
- The conditional probability formula is:
[tex]\[
P(P \mid E) = \frac{P(P \cap E)}{P(E)}
\][/tex]
7. Calculate [tex]\( P(P \mid E) \)[/tex]:
- Substitute [tex]\( P(P \cap E) \)[/tex] and [tex]\( P(E) \)[/tex] into the formula:
[tex]\[
P(P \mid E) = \frac{\frac{1}{6}}{\frac{1}{2}}
\][/tex]
8. Simplify the Expression:
- Dividing by a fraction is the same as multiplying by its reciprocal. Thus:
[tex]\[
P(P \mid E) = \frac{1}{6} \times \frac{2}{1} = \frac{2}{6} = \frac{1}{3}
\][/tex]
Therefore, the probability of rolling a prime number given that the number is even is:
[tex]\[
\boxed{\frac{1}{3}}
\][/tex]
