I've been reviewing prob and stats which I took 10 years ago (ulp!) and came across a problem I can't solve. Consider the problem:(adsbygoogle = window.adsbygoogle || []).push({});

What is the expected minimum number of coin tosses you'd need to make in order to get 3 tails in a row?

Initial thought: the probability of getting a H followed by 3 T is p = (1/2)^4. These are Bernoulli trials, so the expected number of trials to get 3 T is:

E(T=3) = 1/p = 2^4 = 16

However, I've seen that the probability is really p = (1/2)^4 - 2, and the expected number of tosses to get 3 tails is really:

E(T=3) = 2^4 - 2 = 14

I wrote a program in C to calculate this expectation, and the average looks very 14-ish, definitely not 16-ish:

$ ./a.out

Average is 13.992740.

$ ./a.out

Average is 13.996880.

$ ./a.out

Average is 14.003285.

$ ./a.out

Average is 13.956330.

$ ./a.out

Average is 13.982135.

$ ./a.out

Average is 13.982135.

$ ./a.out

Average is 14.000490.

so it appears that 14 is correct. I'm really struggling to understand that "-2" in the probability and expectation value. Does anyone understand it?

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# Homework Help: Expectations of coin tossing

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