Random walk with endpoints

The Riddler post for today is about a bar game where you flip a coin and move forward or backward depending on the result. Here is the problem:

Consider a hot new bar game. It’s played with a coin, between you and a friend, on a number line stretching from negative infinity to positive infinity. (It’s a very, very long bar.) You are assigned a winning number, the negative integer -X, and your friend is assigned his own winning number, a positive integer, +Y. A marker is placed at zero on the number line. Then the coin is repeatedly flipped. Every time the coin lands heads, the marker is moved one integer in a positive direction. Every time the coin lands tails, the marker moves one integer in a negative direction. You win if the coin reaches -X first, while your friend wins if the coin reaches +Y first. (Winner keeps the coin.)

How long can you expect to sit, flipping a coin, at the bar? Put another way, what is the expected number of coin flips in a complete game?

Here is my solution:
[Show Solution]

and here is a much slicker solution, courtesy of Daniel Ross:
[Show Solution]

3 thoughts on “Random walk with endpoints”

  1. Nice post Laurent – although as a committed maths nerd, I was sorry to see the general case of the biased coin relegated to a link! ; )

    If you’re interested, I’ve written up some thoughts on the solution and another Riddler extension on my own blog here: theexcelements.com/

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