Consider a coin that comes up heads with probability P and

Chapter , Problem 40

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QUESTION:

Consider a coin that comes up heads with probability P and tails with probability 1 - p. Let qn be the probability that after n independent tosses, there have been an even number of heads. Derive a recursion that relates qn to qn-I, and solve this recursion to establish the formula qn = (1 + (1 - 2Pt) /2.

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QUESTION:

Consider a coin that comes up heads with probability P and tails with probability 1 - p. Let qn be the probability that after n independent tosses, there have been an even number of heads. Derive a recursion that relates qn to qn-I, and solve this recursion to establish the formula qn = (1 + (1 - 2Pt) /2.

ANSWER:

Step 1 of 2

Assume that

A: Event that initialtosses produce an even number of heads.

E: Event that thetoss results in a head

Obtaining an even number of head in n tosses is possible in two distinct ways:

Case1: The firsttosses results in an even number of heads, and the text, that is, the  toss results in tails: this the event  

Case2: The firsttosses results in an odd number of heads, and the text, that is, the  toss results in heads: this the event  

Again, A and E are the independent events, where

         

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