A die is rolled twice and the sum of the numbers that appear is observed. What is the probability that the sum is either a perfect square or a perfect cube?
To find the probability that the sum of the numbers rolled on a die is either a perfect square or a perfect cube, we can break this down into two cases and then add the probabilities of each case.
Case 1: The sum is a perfect square.
There are 6 faces on a standard die, and the numbers on the faces are 1, 2, 3, 4, 5, and 6. To get a perfect square sum, the possible outcomes are:
1. Rolling a 1 on the first die and a 1 on the second die, which gives a sum of 1 + 1 = 2 (not a perfect square).
2. Rolling a 4 on the first die and a 1 on the second die, which gives a sum of 4 + 1 = 5 (not a perfect square).
3. Rolling a 1 on the first die and a 4 on the second die, which gives a sum of 1 + 4 = 5 (not a perfect square).
4. Rolling a 4 on the first die and a 4 on the second die,
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