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Chapter 8: Problem 5

Let \(X\) denote the random variable that gives the sum of the faces that fall uppermost when two fair dice are rolled. Find \(P(X=7)\).

Short Answer

Expert verified
The probability of the sum of the faces when two fair dice are rolled being 7 is \(\frac{1}{6}\).

Step by step solution

01

List all possible outcomes

There are two dice, each with 6 faces, so there are a total of \(6 \times 6 = 36\) possible outcomes when the dice are rolled. We can represent each outcome as a pair of the numbers appearing on the two dice. The first entry in the pair represents the number on the first die and the second entry in the pair represents the number on the second die. The possible outcomes are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)
02

Count the ways the sum can be equal to 7

Now, we need to identify which pair of numbers have a sum of 7. These pairs are: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) There are 6 pairs in total whose sum is 7.
03

Calculate the probability

Finally, in order to find the probability that \(X = 7\), we need to divide the total number of ways the sum can be 7 by the total number of possible outcomes: \[P(X=7) = \frac{6}{36} = \frac{1}{6}\] Therefore, the probability that the sum of the faces when two fair dice are rolled is 7 is \(\frac{1}{6}\).

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