Going bust question

[Thythe]

You start with $100. A naive (hmm, maybe) friend plays the following game with you. You roll two fair die and if any combination other than 66 comes up, he pays you $10. If the combination of 66 comes up, you pay him 25 cents. It takes 1 minute to play this game and it must be played into eternity. Do you go bust in this game eventually and if so on average how long does it take? Can this question even be solved?

[kyro]

Wait, let me get this straight. 35/36 times you are going to be paid 10 bucks. And 1/36 times you are paying 25 cents? I would say the odds of you going bust are just over 1 in 14400, the odds that you roll (6,6) 400 straight times. Sure I guess it could happen, but you don't "go bust eventually."

[kyro]

Now if you mean if you roll 66 you pay him 10 dollars and if not he pays you .25, then yes you would go bust eventually.

(35/36)*(.25) + (1/36)*(-10) = -5/36

Therefore, you would lose an average of -.138888889 cents per play. It would take 720 plays to amass 100 bucks lost. So, 720 minutes, or 12 hours.

I think

[Thythe]

I meant it the original way you interpreted it. You are right that you would have to lose 400 times consecutively in the beginning to lose. And after just one win you would have to lose 440 times in a row. Since there is always a chance (albeit extremely small) isn't it inevitable that you will lose eventually? Even if the limit of the loss approaches an infinite time?

[BruceZ]

[ QUOTE ]
I meant it the original way you interpreted it. You are right that you would have to lose 400 times consecutively in the beginning to lose. And after just one win you would have to lose 440 times in a row. Since there is always a chance (albeit extremely small) isn't it inevitable that you will lose eventually? Even if the limit of the loss approaches an infinite time?

[/ QUOTE ]

No. As long as you have the advantage, the probability of going bust is always less than 1. Your bankroll increases linearly with time, but the chance of going bust decreases exponentially with the size of your bankroll (much faster). Many people believe that playing long enough will greatly increase your risk of ruin. In fact, once you get beyond a relatively small length of time, your chance of going broke after that is negligible no matter how long you play. Even if you played for $1, the chance of you going broke would be infinitesimal. It would be (1/35)^100 = 3.91E-155. See derivation here (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Board=genpok&Number=791989&F orum=,All_Forums,&Words=&Searchpage=0&Limit=25&Mai n=780427&Search=true&where=&Name=197&daterange=&ne werval=&newertype=&olderval=&oldertype=&bodyprev=# Post791989). You can use the general risk of ruin equation for this exact problem by plugging in your win rate and standard deviation.