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A fair die is rolled. a. If the die is rolled 8 times, what is the probability that the die comes up 6 exactly twice? b. If the die is rolled until the first 6 comes up, find the probability that the first 6 occurs on the 4th roll. c. If the die is rolled until the third 6 comes up, find the probability that the third 6 occurs on the 5th roll.

Solution:-

a. To find the probability that a fair die comes up 6 exactly twice when rolled 8 times, we can use the binomial probability formula. The probability of getting a 6 on a single roll is , and the probability of not getting a 6 is The formula for the probability of exactly k successes in n independent trials is:

Where:

P(X = k) is the probability of getting exactly k successes,