# Random Walker Problem - Help Needed

I need some help solving this problem.

A man is about to perform a random walk. He is standing a distance of 100 units from a wall. In his pocket, he has 10 playing cards: 5 red and 5 black.

He shuffles the cards and draws the top card.

If he draws a red card, he moves 50 units (half the distance from the wall) to the right (away from the wall).

If he draws a black card, he moves 50 units (half the distance from the wall) to the left (towards the wall).

How far from the wall will he be after all 10 cards have been drawn?

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What happens when he hits the wall? When he tries to go past it? (Presumably "nothing" and "he stays put" but it'll be helpful to make this more precise). – Alon Amit Apr 14 '12 at 4:04
Does he move $50$ yards each time, or does he move half the current distance to the wall each time? – Brian M. Scott Apr 14 '12 at 4:08
@AlonAmit: If you move half the distance each time, you will never hit the wall. – Ross Millikan Apr 14 '12 at 4:12
@RossMillikan: sure, I know, but that's not how I interpreted the question, and why I asked for clarification. I thought it's always 50 units and the question is a variation of a random walk with fixed step and an absorbing barrier. – Alon Amit Apr 14 '12 at 15:02
Thanks for your questions and answers. He moves 50 units each time he draws a card. Once he hits the wall, he stays put but keeps drawing cards until all 10 cards are drawn. – Scott Post Apr 14 '12 at 19:59

HINT: If he always moves half the current distance to the wall, his distance from the wall is multiplied either by $\frac12$ or by $\frac32$ each times.