If a standard six-sided die is rolled twice, what is the probability that the first roll shows an even number and the second shows a number greater than two as a reduced fraction?

To determine the probability that the first roll shows an even number and the second roll shows a number greater than two, we first identify the possible outcomes for each die roll.

For the first roll, the even numbers on a six-sided die are 2, 4, and 6. This gives us a total of 3 favorable outcomes (even numbers) out of 6 possible outcomes. The probability that the first roll is even is therefore:

[

P(\text{first roll is even}) = \frac{3}{6} = \frac{1}{2}

]

For the second roll, the numbers greater than two are 3, 4, 5, and 6. This provides us with 4 favorable outcomes out of 6 possible outcomes. The probability that the second roll is greater than two is:

[

P(\text{second roll > 2}) = \frac{4}{6} = \frac{2}{3}

]

Now, we want to find the combined probability of both events happening. Since the rolls of the die are independent events, we can multiply the probabilities of the two events:

[

P(\text{first roll even and second roll > 2})