I don't understand!!! How given (2, aa) answer is 6. I think that must be 4!!!  Please help, who know that!!!!! (Я не понел Как 2 и аа могут дать ответ 6 Я думаю что должно быть 4 Помогите------------)

expected time is inifinite sum of time * p(time), where p(time) is the probability to get resulting string exactly in 'time' steps.
for aa:
probability to get 'aa' in exactly 1 step is 0. (simple)
probability to get 'aa' in exactly 2 steps is 1/4 (simple)
probability to get 'aa' in exactly n steps is probability to get a string of length == n - 1 that ends with 'a', and doesn't contain 'aa' as substring multiplyed by 1/2 (when you have such string there is a 1/2 chance that the next character will be 'a').
for n == 3: 1/8, for n == 4: 1/8 etc.
If you simulate such steps and get a sum for n = [2.. 10000] you will get answer equal to 6

PS.
this is simulation of 10 steps:
0.000000 * 1
0.250000 * 2
0.125000 * 3
0.125000 * 4
0.093750 * 5
0.078125 * 6
0.062500 * 7
0.050781 * 8
0.041016 * 9
0.033203 * 10