Consider a die with eight sides, marked one, two, three, and so on. Assuming equally likely outcomes, find the probability that the sum of two dice is the given number. (Enter the answer either as a fraction or as a decimal rounded to three places.)
(a) P(2) = 1
(b) P(5) = 2
(c) P(7) = 3
(d) P(8) = 4
(e) P(12) = 5
(f) P(15) = 6
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To find the probability of the sum of two dice being a specific number, we need to determine the number of ways that number can be obtained and divide it by the total possible outcomes.
There are 8 sides on each die, so the total number of outcomes is 8 * 8 = 64.
(a) P(2): To get a sum of 2, we need to have both dice show 1. There is only one way to achieve this. So P(2) = 1/64.
(b) P(5): To get a sum of 5, we can have the following combinations: (2, 3), (3, 2), (1, 4), (4, 1). There are 4 ways to obtain a sum of 5. So P(5) = 4/64 = 1/16.
(c) P(7): To get a sum of 7, we can have the following combinations: (1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3). There are 6 ways to obtain a sum of 7. So P(7) = 6/64 = 3/32.
(d) P(8): To get a sum of 8, we can have the following combinations: (2, 6), (6, 2), (3, 5), (5, 3), (4, 4). There are 5 ways to obtain a sum of 8. So P(8) = 5/64.
(e) P(12): To get a sum of 12, we need both dice to show 6. There is only one way to achieve this. So P(12) = 1/64.
(f) P(15): It is not possible to obtain a sum of 15 with two eight-sided dice, so P(15) = 0.
So the probabilities are:
(a) P(2) = 1/64
(b) P(5) = 1/16
(c) P(7) = 3/32
(d) P(8) = 5/64
(e) P(12) = 1/64
(f) P(15) = 0.