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Problem Analysis:

A die is rolled twice, and we need to find the probability of either:

1. Rolling a 5, or
2. Getting an even number.

A die has 6 faces, numbered 1 to 6. Therefore:

• The even numbers are 2, 4, and 6.
• We also want to roll a 5.

Let’s consider the possible outcomes for each roll and determine the total number of successful outcomes.

Total Outcomes:

Since a die is rolled twice, the total number of outcomes is $6 \times 6 = 36$.

Favorable Outcomes:

We can break the favorable outcomes into two categories.

1. Rolling a 5 on either die:

   • For the first roll: (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)
   • For the second roll: (1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5)

   Notice that (5, 5) is repeated, so we must count it only once.

   • Number of favorable outcomes for rolling a 5 = 11

2. Getting an even number:

   • Even numbers are 2, 4, and 6.
   • We can have:
     • (2, _), (4, _), (6, _) for all values from 1 to 6.
     • (, 2), (, 4), (_, 6) for all values from 1 to 6.

   Since we are counting all outcomes that are even on either die, and since there may be overlapping outcomes, we need to count these combinations carefully to avoid double-counting.

Overlapping Cases (Rolling a 5 and an even number):