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 8.Event B: The sum is divisible by 4 or 6 (or both).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 8.Event B: The sum is divisible by 4 or 6 (or both).Write your answers as fractions.(a) PA = (b) PB =
Solution
To solve this problem, we first need to understand the total number of possible outcomes when rolling a die twice. Since a die has 6 faces, the total number of outcomes is 6*6 = 36.
(a) Event A: The sum is greater than 8.
The possible outcomes where the sum is greater than 8 are: (3,6), (4,5), (4,6), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6). That's 10 outcomes. So, the probability of event A, P(A), is the number of favorable outcomes divided by the total number of outcomes.
P(A) = 10/36 = 5/18.
(b) Event B: The sum is divisible by 4 or 6 (or both).
The possible outcomes where the sum is divisible by 4 are: (1,3), (2,2), (3,1), (3,5), (4,4), (5,3), (2,6), (6,2). That's 8 outcomes.
The possible outcomes where the sum is divisible by 6 are: (1,5), (2,4), (3,3), (4,2), (5,1), (6,6). That's 6 outcomes.
However, we have double counted the outcomes where the sum is divisible by both 4 and 6, which are: (2,2), (4,4), (6,6). That's 3 outcomes.
So, the total number of outcomes where the sum is divisible by 4 or 6 is 8 + 6 - 3 = 11.
So, the probability of event B, P(B), is the number of favorable outcomes divided by the total number of outcomes.
P(B) = 11/36.
So, the answers are:
(a) P(A) = 5/18 (b) P(B) = 11/36
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