another probably simple coin toss question

fuzzy · 2005-01-11, 20:06

Alice and Bob play coin toss: Alice pays Bob £1 for each head and Bob pays Alice £1 for each tail they throw. They continue playing until one player loses (runs out of money). Initially Alice has £6 and Bob has £14.

a. Determine, with proof, the probability that Alice loses.
b. Determine the probability that Alice loses but also has at some time previously been within £1 of winning.

again im sure this must be easy for most but probability was never my strong point anyway. its so frustrating not being able to do it!

Orgasmic Troll · 2005-01-12, 02:09

This might have some helpful info:

http://mathworld.wolfram.com/RandomW...mensional.html

wblipp · 2005-01-12, 04:55

The most straight forward approach is to define 21 variables, A_n ranging from A₀ to A₂₀, as the probability that Alice will lose if she "starts" with £n. Clearly A₀=1 and A₂₀=0. For n=1 to 19, Alice with either win or lose the next toss, and then "starts over" from there. Hence
A_n = 0.5 * A_n-1 + 0.5 * A_n+1 for n=1 to 19.

Solve this set of equations, and A₆ is the answer to the first question.

For the second question you could solve a similar problem with only £19 total. And then multiply the probability of Alice winning that game times A₁₉.

There are lots of cute relationships you can find. For example, it you have the probabilities for a game with £m, you can calculate the probabilites for a game with £m+1 by observing that Alice first either wins or loses the £m game and then figuring out the consequences of the next few tosses.

akruppa · 2005-01-12, 08:39

Google for "gambler's ruin"

Alex

2005-01-11, 20:06	#1
fuzzy Jan 2005 2 Posts	another probably simple coin toss question Alice and Bob play coin toss: Alice pays Bob £1 for each head and Bob pays Alice £1 for each tail they throw. They continue playing until one player loses (runs out of money). Initially Alice has £6 and Bob has £14. a. Determine, with proof, the probability that Alice loses. b. Determine the probability that Alice loses but also has at some time previously been within £1 of winning. again im sure this must be easy for most but probability was never my strong point anyway. its so frustrating not being able to do it!

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2005-01-12, 02:09	#2
Orgasmic Troll Cranksta Rap Ayatollah Jul 2003 641 Posts	This might have some helpful info: http://mathworld.wolfram.com/RandomW...mensional.html

2005-01-12, 04:55	#3
wblipp "William" May 2003 New Haven 2·7·13² Posts	The most straight forward approach is to define 21 variables, A_n ranging from A₀ to A₂₀, as the probability that Alice will lose if she "starts" with £n. Clearly A₀=1 and A₂₀=0. For n=1 to 19, Alice with either win or lose the next toss, and then "starts over" from there. Hence A_n = 0.5 * A_n-1 + 0.5 * A_n+1 for n=1 to 19. Solve this set of equations, and A₆ is the answer to the first question. For the second question you could solve a similar problem with only £19 total. And then multiply the probability of Alice winning that game times A₁₉. There are lots of cute relationships you can find. For example, it you have the probabilities for a game with £m, you can calculate the probabilites for a game with £m+1 by observing that Alice first either wins or loses the £m game and then figuring out the consequences of the next few tosses.

2005-01-12, 08:39	#4
akruppa "Nancy" Aug 2002 Alexandria 9A3₁₆ Posts	Google for "gambler's ruin" Alex