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: 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?