Question

An experiment consist of rolling 3 fair dice (1 red die, a blue die, and a white die) and recording the number rolled on each die. Assume that the dice are fair, so that all outcomes are equally likely.

1. What probability should be assigned to each outcome?
2. What is the probability that the sum of the numbers rolled is 6?
3. What is the probability that the sum of the numbers rolled is at most 5?

Answer 1

In 1, there are 6 outcomes for each die, so for three dice, the total combination is 6 x 6 x 6 = 216 outcomes. Hence, the probability of any individual outcome is 1/216

The outcomes that will add up to 6 are
1+1+4
1+4+1
4+1+1
1+2+3
1+3+2
2+1+3
2+3+1
3+1+2
3+2+1
2+2+2

Hence the probability is 10/216

In 3, the minimum sum of the three dice is 3. so we start with this
P(n = 3)
1+1+1 ; 1/216

P(n = 4)
1+1+2
1+2+1
2+1+1 ; 3/216

P(n = 5)
1+1+3
1+3+1
3+1+1
1+2+2
2+1+2
2+2+1; 6/216

The sum in 3 is 10/216 or 5/108