A fair coin is tossed 50 times. The outcomes are written in order, producing a 50-letter “word” consisting of the letters H and T.

Question

Compute the expected number of occurrences of “HHH” in this word (overlaps are allowed). For example, the word THHHHTTHHHTH has 3 such occurrences.

I have that Ai is 1 if HHH occurrence happens at the ith place and 0 otherwise

so $$X=\sum Ai\space from \space i=1 \space to \space 48$$

But I don't know where to go from here.

Answer 1

Hints:

• What is the probability the first three tosses are HHH?
• What is $\mathbb E[A_1]$, the expected value of $A_1$?
• What is $\mathbb E[A_i]$?
• What is $\sum_i \mathbb E[A_i]$?
• What is $\mathbb E\left[\sum_i A_i\right]$, the expected value of $X$?