We roll two fair 6-sided dice, A and B. Each one of the 36 possible outcomes is assumed to be equally likely. 1) Find the probability that dice A is larger than dice B. 2) Given that the roll resulted in a sum of 5 or less, find the conditional probability that the two dice were equal. 3) Given that the two dice land on different numbers, find the conditional probability that the two dice differed by 2.

1) 41.67% probability that dice A is larger than dice B.

2) Given hat the roll resulted in a sum of 5 or less, there is a 20% conditional probability that the two dice were equal.

3) Given that the two dice land on different numbers there is a 26.67% conditional probability that the two dice differed by 2.

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.

1) Find the probability that dice A is larger than dice B.

Desired outcomes:

(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)

There are 15 outcomes in which dice A is larger than dice B.

There are 36 total outcomes.

So there is a 15/36 = 0.4167 = 41.67% probability that dice A is larger than dice B.

2) Given that the roll resulted in a sum of 5 or less, find the conditional probability that the two dice were equal.

Desired outcomes:

Sum of 5 or less and equal

(1,1), (2,2)

There are 2 desired outcomes

Total outcomes:

Sum of 5 or less

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

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

(3,1), (3,2)

(4,1)

There are 10 total outcomes.

So given hat the roll resulted in a sum of 5 or less, there is a 2/10 = 20% conditional probability that the two dice were equal.

3) Given that the two dice land on different numbers, find the conditional probability that the two dice differed by 2.

Desired outcomes

Differed by 2

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

There are 8 total outcomes in which the dices differ by 2.

Total outcomes:

There are 30 outcomes in which the two dice land of different numbers.

So given that the two dice land on different numbers there is a 8/30 = 0.2667 = 26.67% conditional probability that the two dice differed by 2.