Consider the experiment of keeping flipping a fair coin until two heads are flipped consecutively. the outcome is the sequence of the flipped result. possible examples of the outcome include hththh and ttththtthh . the random variable x is defined to be the number of 3 flips of an outcome. for example, x(hththh) = 6 and x(ttththtthh) = 10 . find e(x) . hint: consider three events: f first two flips are , first two 1 = hh f2 = flips are ht , f first flip is . note that , , and form a 3 = t f1 f2 f3 partition of s . the law of total expectation states that e(x) = e(x|f )p(f ) (x|f )p(f ) (x|f )p(f ) , see exercise 1 1 + e 2 2 + e 3 3 7.4.22. try to express e(x|f ) , , and in terms of . 1 e(x|f )2 e(x|f )3 e(x) instead of using the formula of conditional expectation e(x|f) = ∑ (x |f) in exercise 7.4.21, think of as "the r∈x(s) r · p = r e(x|f) expectation of x given that event f occurs."