Question

Suppose you roll three fair six-sided dice and add up the numbers you get. What is the probability that the sum is at least 16?

Answer 1

Solution:

Probability of getting any number (from 1 to 6) on rolling a fair six-sided dice is 1/6.

When we roll three fair six-sided dice, we can get maximum sum of 18.

We have to obtain P(sum = at least 16).

P(sum = at least 16)= P(sum = 16) + P(sum = 17)+ P(sum =18)

When we roll three dice, different ways of obtaining sum of 16 and corresponding probabilities are given below:

Dice 1	Dice 2	Dice 3	Probability
4	6	6	(1/6)×(1/6)×(1/6) = 1/216
6	4	6	(1/6)×(1/6)×(1/6) = 1/216
6	6	4	(1/6)×(1/6)×(1/6) = 1/216
5	5	6	(1/6)×(1/6)×(1/6) = 1/216
5	6	5	(1/6)×(1/6)×(1/6) = 1/216
6	5	5	(1/6)×(1/6)×(1/6) = 1/216

Hence, P(sum = 16) = [(1/216) + (1/216) + (1/216) +(1/216) + (1/216) + (1/216)] = 6/216

When we roll three dice, different ways of obtaining sum of 17 and corresponding probabilities are given below:

Dice1	Dice 2	Dice 3	Probability
5	6	6	(1/6)×(1/6)×(1/6) = 1/216
6	5	6	(1/6)×(1/6)×(1/6) = 1/216
6	6	5	(1/6)×(1/6)×(1/6) = 1/216

Hence, P(sum = 17) = [(1/216) + (1/216) + (1/216)] = 3/216

When we roll three dice, different ways of obtaining sum of 18 and corresponding probabilities are given below:

Dice1	Dice 2	Dice 3	Probability
6	6	6	(1/6)×(1/6)×(1/6) = 1/216

Hence, P(sum = 18) = 1/216

Hence, P(sum = at least 16) = (6/216) + (3/216) + (1/216)

(sum = at least 16) = 10/216 = 0.0462963

On rolling three fair six-sided dice, the probability that the sum is at least 16, is 10/216 = 0.0462963.

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