InterviewSolution
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InterviewSolution
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It is tossed n times. Let `P_n` denote the probability that no two (or more) consecutive heads occur. Prove that `P_1 = 1,P_2 = 1 - p^2 and P_n= (1 - P) P_(n-1) + p(1 - P) P_(n-2)` for all `n leq 3`. |
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Answer» Given that the probability of showing head by a coin when tossed is p. So, the probability of coin not showing head is (1-p). Now, `p_(n)` denotes probability that no two or more consecutive heads occur in n throws. Clearly, `p_(1)=1` as when coin is tossed once there will be no two consectiven heads. Also, `p_(2)=P(HT)+P(TH)+P(T T)` `=p(1-p)+p(1-p)+(1-p)^(2)=1-p^(2)` Let event A is "last toss is tail" and evetn B is "last toss is head and second last toss is tail." `therefore` Using total probability theorem, `p_(n)=p_(n-1)xxP(A)+p_(n-2)xxP(B)` `thereforep_(n)=(1-p)p_(n-1)+p(1-p)p_(n-2)"for all n"ge3.` |
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