Four fair six-sided dice are rolled. The probability that the sum of the…
2014
Four fair six-sided dice are rolled. The probability that the sum of the results being 22 is \(X / 1296\). The value of \(X\) is _______
Attempted by 57 students.
Show answer & explanation
Correct answer: 10
We need the number of outcomes where four fair six-sided dice (values 1–6) sum to 22, out of a total of 6^4 outcomes.
Total possible outcomes: 6^4 = 1296
Let the four dice results be x1, x2, x3, x4 with 1 ≤ xi ≤ 6. Define yi = 6 − xi, so 0 ≤ yi ≤ 5 and y1 + y2 + y3 + y4 = 24 − 22 = 2.
Count nonnegative integer solutions to y1 + y2 + y3 + y4 = 2. Using stars and bars, the number is C(5,3) = 10. (The upper bounds yi ≤ 5 are not restrictive here because the total 2 ≤ 5.)
Therefore the number of favorable outcomes X = 10, and the probability is 10/1296.
Answer: 10