## Tuesday, June 07, 2011

### Random Walk

A drunk leaves a bar (rather forcefully).  After standing up and dusting himself off (rather unsuccessfully), he decides to walk home.  Unfortunately for him, he has no idea where home is and has no way of deciding which way to go.  He is on a street traveling east-west only, making things simpler for him.

With each step, he's just as likely to turn around and walk back in the direction he came from.  After 20 steps, how likely is it he will be back where he started from (right in front of the bar)?

1. (1/2)^20 = (1^20/2^20) = 1/1048576 assuming he's not already taken one step east-west, otherwise it's (1/2)^19 = 1/524288

2. 50/50. He either is, or is not, back where he started from.

3. This is like flipping a coin 20 times. You need to figure out what the probability of getting 10 heads and 10 tails is (taking 10 steps to the right and 10 steps to the left. So, endothief has the right idea, but hasn't finished the problem.

Seems counter-intuitive, doesn't it?

4. 0.176197052
184756 of 1048576

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