Problem 5

The Bank of Bath issues coins with an

H

on one side and a

T

on the other. Harry has

n

of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly

k>0

coins showing

H

, then he turns over the

k

th coin from the left; otherwise all coins show

T

and he stops. For example, if

n=3

, the process starting with

THT

THT\to HHT\to HTT\to TTT

, which stops after three operations. (a) Show that, for each initial configuration, Harry stops after a finite number of operations. (b) For each initial configuration

C

, let

L(C)

be the number of operations before Harry stops. Determine the average value of

L(C)

over all

2^n

possible initial configurations.

Step 1 of 6: Define the expected value

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In plain words

We prove termination and compute the average together.

E_n=\mathbb E[L(C)]

Detailed analysis

Let $E_n$ be the average number of operations for $n$ coins. We use induction on $n$ . For $n=1$ , the configurations $T$ and $H$ take respectively $0$ and $1$ operations, so the process terminates and $E_1=\frac12$ .