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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 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 < k <n are independent (This requires you to show that PEP(Ek) for every choice of n-nand ki 〈 . . . 〈km) (b) Show that if the coin is not fair, the events are not independent. (An unfair coin gives H with probability pメ

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