A pair of fair dice is rolled. (a) What is the probability that both dice show the same number? (b) What is the probability that both dice show different numbers? (c) What is the probability that the second die lands on a lower value than does the first?

a) 16.7% probability that both dice show the same number

b) 83.3% probability that both dice show different numbers

c) 41.67% probability that the second die lands on a lower value than does the first.

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

In this problem, we have these possible outcomes:

Format (Dice A, Dice B)

(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)

There are 36 possible outcomes.

(a) What is the probability that both dice show the same number?

(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)

6 outcomes in which both dice show the same number.

6/36 = 0.167

16.7% probability that both dice show the same number

(b) What is the probability that both dice show different numbers?

The other 30 outcomes

30/36 = 0.833

83.3% probability that both dice show different numbers

(c) What is the probability that the second die lands on a lower value than does the first?

(2,1)

(3,1), (3,2)

(4,1), (4,2), (4,3)

(5,1), (5,2), (5,3), (5,4)

(6,1), (6,2), (6,3), (6,4), (6,5)

15 outcomes in which the second die lands on a lower value than does the first.

15/36 = 0.4167

41.67% probability that the second die lands on a lower value than does the first.