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# Bet you a dollar

## Question

Starting with 0 dollars, you repeatedly flip a fair coin, earning \$1 each time heads appears, and losing \$1 each time tails appears. When your net cash reaches either A or -B, you stop gambling.
As a function of A and B, compute the expected number of flips until the game stops.
Now consider the unstopped version of this process, in which you gamble indefinitely regardless of your current profit or debt. Prove that the expected time till your net cash is +\$1 is infinite. Likewise, the expected time till your net cash is -\$1 is also infinite.
note that one of these two events must occur upon the first coin flip!

## 1 answer to this question

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part 1) 2^((A-B)/2)

part 2) is not expected time of inifinity till net cash is 1\$.

as you say, you get net cash 1\$ on your first flip. the expected number of flips untill you get a specific value is 2^A. if you play the game indefinitely, all values are equally likely.

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