Question

Let N = the number of the tossing a coin when we get the first HEAD. M = the number of the tossing a coin when we get the second HEAD. We assume probability of getting a HEAD is p. Find the Probability Distribution of N and M and are they independent?

Answer 1

P(M) = P(tails in first M-1 tooses)*P(heads in Mth toss) = [P(tails)^(M-1)]*P(heads)

= [(1/2)^(m-1)] * (1/2)

= (1/2)^m

P(N) = P(1 head in N-1 tosses)*P(heads in Nth toss)

: use to calculate P(1 head in n-1 tosses)

= [((n-1)C1)*(1/2^1)*(1/2^(n-1))] * (1/2)

= [((n-1))*((1/2)^1)*((1/2)^(n-1))] * (1/2)

P(N) = [(N-1)*((1/2)^N)]*(1/2)

P(N) depends on M lets say M=5 then P(N=4)=0 {if first heads comes at 5 th toss then second toss can never come at 4th toss}

but if M=1 then P(N=4) > 0

so, we can see they are not independent

P.S. (please upvote if you find the answer satisfactory)