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 of the values rolled on four 6-sided dice is greater than or equal to k is P(X≥k) = ((7-k)^4) / 6^4, for k between 1 and 6.

Step-by-step explanation:

The student is asking about the probability that the minimum of the values rolled with four 6-sided dice is greater than or equal to a particular number k. To find the probability P(X≥k), we must first determine the number of possible outcomes where each die rolls a value of k or higher. Since the dice are independent, the probability that a single die is k or greater is (7-k)/6 because there are 6-k+1 outcomes from k to 6 that satisfy this condition for one die.

For all four dice to have a value of at least k, we must multiply the individual probabilities. Therefore, P(X≥k) for one die is ((7-k)/6) to the fourth power since we want this outcome for all four dice. So, P(X≥k) = ((7-k)^4) / 6^4. Note that since the die only has values from 1 to 6, this formula is valid for k between 1 and 6 (inclusive). For k less than 1, P(X≥k) is 1, as the minimum value rolled will always be at least 1, and for k greater than 6, P(X≥k) is 0, as it is impossible for a minimum value of a die to exceed 6.