Let Qn denote the probability that in n tosses of a fair coin no run of 3 consecutive heads appears. Show that:
Qn = ½ Qn-1 + ¼Qn-2 + ⅛Qn-3
Q0 = Q1 = Q2 = 1
HINT: Condition of the first tail.
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First, let us consider first three tosses in n tosses. There will be no 3 consecutive heads appear if and only if:
CASE #1: If the first coin toss results T, there must be no 3 consecutive heads appear in the next n - 1 tosses.
CASE #2: If the first coin toss results H, then the next toss must result T and there must be no 3 consecutive heads appear in the next n - 2 tosses.
CASE #3: If the first two coin tosses result HH, ...
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