Question

Four 6-sided dice are rolled. The dice are fair, so each one has equal probability of producing a value in {1, 2, 3, 4, 5, 6}. Let X = the minimum of the four values rolled. (It is fine if more than one of the dice has the minimal value.). What is P (X ≥ k) as a function of k?

Answer 1

The probability that the minimum value rolled, X, is greater than or equal to k can be determined by analyzing the complementary event: the probability that X is less than k. The probability that X is less than k can be calculated by finding the probability that all four dice rolls are less than k. Therefore, P(X ≥ k) = 1 - ((k-1)/6)^4.

Step-by-step explanation:

The probability that the minimum value rolled, X, is greater than or equal to k can be determined by analyzing the complementary event: the probability that X is less than k.

The probability that X is less than k can be calculated by finding the probability that all four dice rolls are less than k. Since each die roll is independent and equally likely to produce a value in {1, 2, 3, 4, 5, 6}, the probability that a single die roll is less than k is (k-1)/6. Therefore, the probability that all four dice rolls are less than k is ((k-1)/6)^4.

Therefore, P(X ≥ k) = 1 - ((k-1)/6)^4.