Question

Problem 2. Consider n flips of a coin. A run is a sequence of consecutive tosses with the same result. For k 〈 n, let Ek be the event that a run is completed at time k; this means that the results of the kth and k1)st flips are different. For example, if n 10 and the outcomes of the first 10 flips are HHHTTHHTTH then runs are completed at times 3, 5,7,9 (a) Show that if the coin is fair, then the events Ek, 1 Sk < n are independent (This requires you to show thatPEP ER for every choice of m < n and ki < < km) (b) Show that if the coin is not fair, the events are not independent. (An unfair coin gives H with probability p^ j.)

Answer 1

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