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Л blilscd coun is toss(xl repcaixlly. I:ach timc thexe Is a probability p (1 a head ino uti up. What is the probability that an even number of heads has occurred after n tosses. (Hint: zero is an even number, and let pn be the probability that an even number of heads has occurred after n tosses)

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Answer #1

Here we want to find the probability that an even number of heads has occurred after n tosses.

Let X = Number of heads occurred after n tosses.

Therefore, X follows binomial distribution with probability of getting head = p and total number of trials = n

For n = even, we have

Required probability = P(X = 0) + P(X = 2) + P(X = 4) + .....+P(X = n -2 ) + P(X = n)

=sum_{i=0}^{n/2}inom{n}{2i}p^{2i}(1-p)^{n-2i}

For n = odd, we have.

Required probability = P(X = 0) + P(X = 2) + P(X = 4) + .....+P(X = n -3 ) + P(X = n - 1)

n-22 2i

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