# Probability Question?

Two players A & B roll a fair die simultaneously, say player A gets x & player B gets y. Only the player with greater number will move |x-y| steps ahead (in case of x=y, no one will move).Currently Player A is 6 steps ahead of player B but 18 steps away from the finishing point, find the probability of Player B...
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Two players A & B roll a fair die simultaneously, say player A gets x & player B gets y. Only the player with greater number will move |x-y| steps ahead (in case of x=y, no one will move).Currently Player A is 6 steps ahead of player B but 18 steps away from the finishing point, find the probability of Player B wining the game.

Update:
Thanks for all the answer so far. I was out of town so could not communicate.
@Todd ,Mathman & Michael:
Yes your results are matching with mine. In fact I did this on Pari/GP & Mathematica and with about 10^9 run I got probability very close to 30% for B with average 19.98 rolls.
As Mathman rightly said...
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Thanks for all the answer so far. I was out of town so could not communicate.

@Todd ,Mathman & Michael:

Yes your results are matching with mine. In fact I did this on Pari/GP & Mathematica and with about 10^9 run I got probability very close to 30% for B with average 19.98 rolls.

As Mathman rightly said I also thought it should take more rolls based on average advance. That's why I suspected my result and wanted to verify. With such clean answer (approaching 30%) I am sure analytical solution must exist, but not sure how to proceed.

@Todd ,Mathman & Michael:

Yes your results are matching with mine. In fact I did this on Pari/GP & Mathematica and with about 10^9 run I got probability very close to 30% for B with average 19.98 rolls.

As Mathman rightly said I also thought it should take more rolls based on average advance. That's why I suspected my result and wanted to verify. With such clean answer (approaching 30%) I am sure analytical solution must exist, but not sure how to proceed.

Update 2:
Todd: I am not sure how Discrete Time Markov Chain can help in this question

Update 3:
@Todd: As a general remark, as far as I know Excel 2003 onward the rand() implementation has passed Diehard test & NIST test but It has not yet passed TestU01 Crush test. Thus it is much better now and will give reliable results for 10^13 random sequences.

Update 4:
Gianlino: If you notice my comment "With such clean answer (approaching 30%) I am sure analytical solution must exist, but not sure how to proceed." indicates that I would like to know analytical solution which only math section can give, sometimes bottom up approach works the best along with analytical...
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Gianlino: If you notice my comment "With such clean answer (approaching 30%) I am sure analytical solution must exist, but not sure how to proceed." indicates that I would like to know analytical solution which only math section can give, sometimes bottom up approach works the best along with analytical thinking.

Here is the best example http://answers.yahoo.com/question/index?...

Here is the best example http://answers.yahoo.com/question/index?...

Update 5:
Gianlino that's what I meant by combined approach. Thanks setting up markov chain
Vasek: Thanks for your programmatic implementation. (I made typo in my earlier comment where I noted 19.98 as avg rolls it was 18.98. I think noise in my answer was different due to excessive number of trials)
I doubted...
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Gianlino that's what I meant by combined approach. Thanks setting up markov chain

Vasek: Thanks for your programmatic implementation. (I made typo in my earlier comment where I noted 19.98 as avg rolls it was 18.98. I think noise in my answer was different due to excessive number of trials)

I doubted my answer more because , I could not believe that on an average Player B would require only additional 2 steps.

Vasek: Thanks for your programmatic implementation. (I made typo in my earlier comment where I noted 19.98 as avg rolls it was 18.98. I think noise in my answer was different due to excessive number of trials)

I doubted my answer more because , I could not believe that on an average Player B would require only additional 2 steps.

Update 6:
Great contributions so far, will keep this open for a day, just in case some one has something more to share.

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