The geometric distributions. You are tossing a balanced die that hasprobability 1/6 of

Chapter 0, Problem 12.47

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The geometric distributions. You are tossing a balanced die that hasprobability 1/6 of coming up 1 on each toss. Tosses are independent. We areinterested in how long we must wait to get the first 1.(a) The probability of a 1 on the first toss is 1/6. What is the probability that thefirst toss is not a 1 and the second toss is a 1?(b) What is the probability that the first two tosses are not 1s and the third toss isa 1? This is the probability that the first 1 occurs on the third toss.(c) Now you see the pattern. What is the probability that the first 1 occurs on thefourth toss? On the fifth toss? Give the general result: what is the probability thatthe first 1 occurs on the kth toss?Comment: The distribution of the number of trials to the first success is called ageometric distribution. In this problem you have found geometric distributionprobabilities when the probability of a success on each trial is 1/6. The same ideaworks for any probability of success.

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