Coin Flipping Expected Value

Question

Find the expected value for the number of flips you'll need to make in order to see the pattern TXT, where T is tails, and X is either heads or tails.

I tried conditioning on the coin flips (e.g. TXT, TXH, H) but I got an incorrect answer of 8. Any suggestions?

Answer 1

I suspect you had TTH happening with probability $\frac{1}{8}$ and giving $E+3$ flips, where $E$ is the expected value of the required number of flips. This is wrong, because we don't have to restart when we get TTH.

TTH leads to TTHT and TTHH with equal probability. TTHT gives length $4$; TTHH gives length $E+4$.

In summary: \begin{align} H: &\frac{1}{2} & (E+1) \\ THT: &\frac{1}{8} & 3 \\ THH: &\frac{1}{8} & (E+3) \\ TTT: &\frac{1}{8} & 3 \\ TTHT: &\frac{1}{16} & 4 \\ TTHH: &\frac{1}{16} & (E+4) \end{align}

Summing them all up and equating to $E$, and solving for $E$ gives $E = \frac{34}{5}$.

Answer 2

Try to find the expected value for the number of flips until TT and then add 1 because X can just be ignored otherwise.