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We have a biased coin, P(heads) = p, X is the number of times we throw is before we see at least two heads and one tail in any order (and anywhere).

I know how to find the PMF for just one head or one tail. It would be P(X = n) = $p^n$ where n is 1, 2, 3, 4, ... and 0 otherwise.

I think I know how to find one head and one tail but I'm not sure. I think it would be P(X = n) = $p^n * (1-p)^n$ where n is 2, 3, 4... and 0 otherwise?

I get stuck with the order issue of the order when I need two heads and one tail. I could see something like $<h,t^x,h>$ or $<h^x,t>$ or $<t^x,h,t^x,h>$ where x is any number > 0.

For example if I got h,h,h,h,h,h,h,h,t I would stop after 9 tosses but if I saw t,h,h,h,h,h,h,h,h I would have stopped after 3 tosses and never seen the rest of the flips.

What I don't know is how to include this idea of order into the PMF.

Thank you!

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  • $\begingroup$ Do we want the last three tosses to contain $2$ heads and $1$ tail (with this not happening earlier) or are two heads and one tail anywhere enough? For example, if we get HTTH do we stop or must we continue? $\endgroup$ – André Nicolas Feb 17 '16 at 17:30
  • $\begingroup$ Two heads and one tails anywhere count $\endgroup$ – Indigo Feb 17 '16 at 18:14
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We condition on the result of the first two tosses.

If we get HT or TH (probability $2p(1-p)$) we are waiting for the third head. The probability that the additional waiting time is $k$ is $(1-p)^{k-1}p$. So in that case we have $\Pr(X=n)=(1-p)^{n-3}p$. That gives a contribution of $$2p(1-p)(1-p)^{n-3}p$$ to the pmf.

If we get HH, (probability $p^2$) the contribution to the pmf is, by a similar argument, $$p^2p^{n-3}(1-p).$$ Finally, we deal with the case TT. We are then waiting for two heads. The probability that this takes exactly $k$ additional trials is the probability of one head in the first $k-1$ additional trials, and head on the $k$-th. The probability of this is $\binom{k-1}{1}p^1(1-p)^{k-2}p$, for a total contribution to the pmf of $$(1-p)^2 \binom{n-3}{1}p(1-p)^{n-4}p.$$ Add up and, if desired, simplify.

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  • $\begingroup$ I understand all but the n-3 choose 1 on the last case. This term grows rather than decreases with n which goes against my intuition that a larger number of flips is less likely. $\endgroup$ – Indigo Feb 17 '16 at 22:02
  • $\begingroup$ Indeed this part of the expression grows. The term $(1-p)^{n-4}$ decreases, in the long run faster than $n-3$ grows, so your general intuition is perhaps not badly violated. We need a total of two heads. If heads are fairly unlikely ($1-p$ close to $1$), it is not unreasonable that we will likely have to wait a while. Here I used the special case waiting time until the second "success" in the negative binomial distribution. $\endgroup$ – André Nicolas Feb 17 '16 at 22:12
  • $\begingroup$ Ahh I see, I was thinking it was n! vs p ^ n but the n! cancels out to just n so it works. Thanks! $\endgroup$ – Indigo Feb 17 '16 at 23:20
  • $\begingroup$ You are welcome. $\binom{n-3}{1}$ is just plain old $n-3$, but I wrote it in binomial coefficient form to give the "source" of the formula. That is also why I did not do the obvious simplifications. $\endgroup$ – André Nicolas Feb 17 '16 at 23:24

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