A die is rolled 2 times. Determine the probability that the sum of the faces is: a) 3 b) 7 c) 10

To find the probabilities for the sum of the faces when a die is rolled 2 times, we will first find the total number of possible outcomes and then calculate the number of favorable outcomes for each sum.

Total number of outcomes when a die is rolled = 6\times6 = 36

a) The sum of the faces is 3:
There are two ways to get a sum of 3: (1, 2) and (2,1)
Probability = \frac{2}{36}=\frac{1}{18}

b) The sum of the faces is 7:
There are six ways to get a sum of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)
Probability = \frac{6}{36} = \frac{1}{6}

c) The sum of the faces is 10:
There are three ways to get a sum of 10: (4, 6) , (5, 5) , (6, 4)
Probability = \frac{3}{36}=\frac{1}{12}

Therefore,
a) P(\text{sum is 3})=\frac{1}{18}
b) P(\text{sum is 7}) = \frac{1}{6}
c) P(\text{sum is 10})=\frac{1}{12}