You toss a fair coin repeatedly until some event happens.
(a) What is the expected number of tosses until the pattern HT appears for the first time? Hint: break this calculation into 2 pieces: the expected amount of time you have to wait until H appears, and the amount of time you have to wait until T appeasr after the first H.
(b) What is the expected number of tosses until the pattern HH appears for the first time? Take a similar approach to part (a), but realize it's not quite as simple.
(c) Notice that the expectation is not the same for these two events. How do we reconcile this with the fact that in two tosses of the coin, HH and HT both have a 1/4 chance of appearing? Why aren't the average waiting times the same by symmetry? Drawing out a sequence of H's and T's might help.