Why does HT come faster than HH?

Question

You flip a fair coin repeatedly and wait for a two-flip pattern to appear.

What is the expected number of flips to first see the pattern HT (heads then tails), and how does it compare to HH (heads then heads)? Explain the difference.

Accepted Answer

HT. Use states by progress. Let E0 be the expected flips with no usable head yet, and E1 once you have a head and are waiting for a tail. E0 = 1 + 	frac12 E1 + 	frac12 E0, \qquad E1 = 1 + 	frac12\cdot 0 + 	frac12 E1. The second equation: after a head, a tail finishes, but another head just keeps you in state 1 (you still have a usable head). So 	frac12 E1 = 1 \Rightarrow E1 = 2. The first gives E0 = 2 + E1 = 4. HT takes 4 flips on average. HH. Same states, but now from state 1 a tail destroys your progress and sends you back to state 0: E1 = 1 + 	frac12\cdot 0 + 	frac12 E0. Combined wi…

Why does HT come faster than HH?

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