EV: 10 Heads in a Row

What is the expected number of flips required to obtain 10 consecutive heads when flipping a fair coin?

Answer

Let $$E_n$$ be the expected number of additional flips needed to reach 10 consecutive heads, given that we currently have $$n$$ consecutive heads. We want $$E_0$$ , the expected flips starting with 0 heads in a row. Clearly,

E_{10} = 0 \quad\text{(since once we have 10 heads, we're done).}

For $$n < 10$$ , on each flip:

- With probability $\tfrac12$ , we flip a head and move from $$n$$ heads to $$n+1$$ heads.

- With probability $\tfrac12$ , we flip a tail and break the streak back to 0 consecutive heads.

Hence, for $0 \le n < 10$ ,

E_n = 1 \;+\; \tfrac{1}{2}E_{n+1} \;+\; \tfrac{1}{2}E_0.

(The extra 1 accounts for the flip we just performed.)

Solving this system step by step or by a standard formula for this type of recurrence shows that $E_0 = 2^{10+1} \;-\; 2 = 2046.$