10 Consecutive Heads

Question

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

QuantSoc · Accepted Answer

Let $E_n$ represent the expected number of additional tosses needed to obtain 10 consecutive heads, given that we already have $n$ consecutive heads. Our goal is to determine $E_0$.

The recurrence relations are:

$$
E_{10} = 0
$$

since no more tosses are needed after obtaining 10 consecutive heads.

For $n < 10$:
- with probability $\frac{1}{2}$, the next toss is heads, moving us to state $E_{n+1}$,
- with probability $\frac{1}{2}$, the next toss is tails, sending us back to state $E_0$.

Thus,

$$
E_n = 1 + \frac{1}{2}E_{n+1} + \frac{1}{2}E_0
$$

Solving the recurrence yields:

$$
E_0 = 2046
$$

Therefore, the expected number of tosses required to obtain 10 consecutive heads is:

$$
\boxed{2046}
$$