Walt sent me this puzzle (reworded from this set of great puzzles from Communications of the ACM; you can get it by accessing from a University or library).
Alice and Bob roll 2 standard 6-sided dice, note their sum, and repeat.
Alice wins if a 7 is rolled, and then followed immediately by another 7.
Bob wins if an 8 is followed immediately by a 7. They continue rolling until somebody wins. Who has the better odds of winning?
Of course the answer is the non-intuitive one. Can you figure out why? As a person with a long-time fondness for craps (I know lots of people who are no good at probability, except when it involves two dice), this seems to me like the ultimate sucker bet to offer someone.
More discussion (and a link to three ways to think about the answer) on JD's notebook.
Does the 8 have to follow the 7 immediately?
ReplyDeleteNot that kind of trick :-). Yes, Bob wins if there is an 8 followed immediately by a 7, and Alice wins if there is a 7 followed immediately by a 7 (there's a typo in your question -- the numbers are backward).
DeleteAnd my apologies for not making the question more clear! A bad start to my blog, and I'll do better in future.
DeleteI'm going to be thinking about this incorrectly ( which is why these types of questions always come with unexpected solutions ), but...
ReplyDelete- Assume the two rolls are IID
- Seven is more likely than eight for the first roll
- Second roll, they both need the same, we can just ignore; even if we don't ignore it, it's equally likely for both of them
- Assume that, if we don't get either a (7, 7) or (8, 7), then neither win
So... Beer is on Alice ?
Sims confirm that Alice wins 2.79% of the time, and Bob only gets there 2.31% of the time.
Your math argument is the standard (wrong) one that makes this a trick question. Your sims need to simulate a whole string of rolls, not just the first two. Hard to imagine how it can be different, which is of course the point.
DeleteFollowup based on Tom's comment : if they're allowed to keep rolling until one of them wins, keeping the last 7 or 8 into account, then it turns out Bob wins.
ReplyDeleteIntuition :
If a 7 is rolled, that can be a win for both Alice and Bob, depending on the last 7 or 8. If it was a 7, Alice wins. If it's an 8, Bob wins. However, for any given roll of an 8, the next 7 cannot allow Alice to win, but Bob will win.
Bob wins about 2.35x as frequently as Alice. Well played, Bob. Well played.
Please see my apology to Tom.
DeletePlease see http://lalashan.mcmaster.ca/theobio/math/index.php/Sucker_bet for more discussion (and a link to the answer).
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