An ordinary (fair) die is a cube with the numbers 1 through 6 on the sides (represented by painted spots). Imagine that such a die is rolled twice in succession and that the face values of the two rolls are added together. This sum is recorded as the outcome of a single trial of a random experiment.Compute the probability of each of the following events.Event A: The sum is greater than 9.Event B: The sum is divisible by 2.Write your answers as fractions.(a) PA = (b) PB =
Question
An ordinary (fair) die is a cube with the numbers 1 through 6 on the sides (represented by painted spots). Imagine that such a die is rolled twice in succession and that the face values of the two rolls are added together. This sum is recorded as the outcome of a single trial of a random experiment.Compute the probability of each of the following events.Event A: The sum is greater than 9.Event B: The sum is divisible by 2.Write your answers as fractions.(a) PA = (b) PB =
Solution
To solve this problem, we need to understand the total number of possible outcomes and the number of outcomes that satisfy the conditions of each event.
(a) Event A: The sum is greater than 9.
The possible sums that are greater than 9 are 10, 11, and 12. Here are the combinations that yield these sums:
- For sum of 10: (4,6), (5,5), (6,4)
- For sum of 11: (5,6), (6,5)
- For sum of 12: (6,6)
So, there are 7 outcomes where the sum is greater than 9.
The total number of outcomes when two dice are rolled is 6*6 = 36 (each die has 6 faces and we are rolling two dice).
So, the probability of Event A, P(A), is the number of favorable outcomes divided by the total number of outcomes, which is 7/36.
(b) Event B: The sum is divisible by 2.
The possible sums that are divisible by 2 are 2, 4, 6, 8, 10, and 12. Here are the combinations that yield these sums:
- For sum of 2: (1,1)
- For sum of 4: (1,3), (2,2), (3,1)
- For sum of 6: (1,5), (2,4), (3,3), (4,2), (5,1)
- For sum of 8: (2,6), (3,5), (4,4), (5,3), (6,2)
- For sum of 10: (4,6), (5,5), (6,4)
- For sum of 12: (6,6)
So, there are 18 outcomes where the sum is divisible by 2.
So, the probability of Event B, P(B), is the number of favorable outcomes divided by the total number of outcomes, which is 18/36 = 1/2.
So, the answers are:
(a) P(A) = 7/36 (b) P(B) = 1/2
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