Show formally that given k successes in n Bernoulli trials, the probability of success for any single trial is k/n. (Hint: Count the number of ways that a success can occur on a particular trial.)
(a) What is the 10 probability of 10 straight successes in a sequence of Bernoulli trials?
(b) Given 10 straight successes, what is the probability of a success on the 11th trial?
(c) The popular conception of the “law of averages” states that the portion of successes in n trials should be P. Show that this can be stated precisely using Chebyshev inequality and is the form of the strong law of large numbers. Reconcile (b) and (c).
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