David Spence: Off-topic, difficult question for Don

[David Spence]

The following question was taken from Peter Griffin's fascinating book, Extra Stuff, and was first posed by Stewart Ethier. I've struggled with it for awhile, and, even though it's about craps instead of blackjack, I am hoping that resident probability expert Don can lend his able mind:

You bet one unit on the pass line or on come on every roll of the dice. Consider your (conditional) expected win on the nth bet, given that you (eventually) lose the first n-1 bets. Show that this tend to a positive limit as n tends to infinity and find it. What is the smallest n for which it is positive? Assume that unresolved come bets are not "off" on come-out rolls.

(I've spent the last two days fumbling with this, to no real avail. By the way, lest anyone thinks this will lead to a winning craps system by just waiting until you lose n bets, consider the important fact that you may not know whether you've lost a bet until some number of rolls after the bet is made. The fact that this problem is conditional upon a FUTURE, instead of a past, roll is what makes it so difficult (and unprofitable :-) )).

David Spence

[S Ethier]

Don S. brought this question to my attention. I'd completely forgotten about the problem and had to think it through again. Here's the intuitive explanation, which requires some justification (left to the reader!).

If the first n-1 rolls are (eventually) lost, there can be no 7s or 11s and no point number can be repeated in the first n-1 rolls. This means that at most 6 point numbers can occur, and for large n all six will occur exactly once (other possibilities are asymptotically negligible). Hence for the nth roll to produce a winner, it must be a 7 or 11. If it's a 7, then all previous bets are lost automatically, whereas if it's an 11, a 7 must still be rolled before any point number. Let p_j be the probability that a total of j is rolled. Then a=p_7/(p_7+p_4+p_5+p_6+p_8+p_9+p_{10})=1/5 is the probability of rolling a 7 before any point number, and the (asymptotic) conditional probability we seek is (p_7+p_{11}a)/a=7/8, hence the (asymptotic) conditional expectation is +3/4.

The second part of the question (what is the smallest n for which it is positive?) should be relatively straightforward once you understand the first part.

[Don Schlesinger]

> Don S. brought this question to my attention. I'd
> completely forgotten about the problem and had to
> think it through again. Here's the intuitive
> explanation, which requires some justification (left
> to the reader!).

Thanks, Stewart, for your swift response. Nice to see you here!

Don

[David Spence]

Thank you for revisiting a problem you created way back in 1985. Your intuitive explanation makes much more sense, and is much more workable, than my caveman-like (apologies to the gentleman in those Geico commercials) process of going through n=1, n=2, etc... and trying to recognize a pattern to find the limit.

Thanks also to Don for bringing this problem to the attention of (clearly) the right person. This forum never ceases to amaze me.