You have a four sided die with 1, 2, 3, 4 spots. What is the sample space when we count the total sum number of spots on the bottom face for rolling the die twice (spots on the first and second rolls)? what is the assignment of probabilites to outcomes
in the sample space? Assume the die is perfectly balanced. What is the probability of 5 spots.

The following are the possible outcomes and their sums

Roll 1

|| 1 | 2 | 3 | 4

======================

1 || 2 | 3 | 4 | 5

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2 || 3 | 4 | 5 | 6

Roll 2 ---------------------------------------

3 || 4 | 5 | 6 | 7

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4 || 5 | 6 | 7 | 8

From this table we see that the sample space is {2,3,4,5,6,7,8}

Also, if the die is fair on every roll then each combination has a 1/16 chance of happening. Therefore you just need to count the number of combinations that give each sum. You should get the following probabilities:

P(S=2) = 1/16

P(S=3) = 2/16 = 1/8

P(S=4) = 3/16

P(S=5) = 4/16 = 1/4

P(S=6) = 3/16

P(S=7) = 2/16 = 1/8

P(S=8) = 1/16