# Sucker bet

### From Math

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.

## Demonstration

The true probability of Alice winning is 47.0%.

The R script Loading WorkingWiki file "alice.R" dynamically. If it doesn't load, click to view the page statically. simulates 50000 trials and gets 47.0% also.

Loading WorkingWiki file "alice.Routput" dynamically. If it doesn't load, click to view the page statically.

Click below for the exact answer.

## Answers

I've written up some ways to think about the answer at Sucker bet/Answer, if you're sure you're ready to look. Here's an excerpt that you might want to think about, before clicking:

*If we look at all consecutive pairs, the naive probabilities must apply: on average 6 out of every 216 pairs must be 7-7 and 5 out of every 216 must be 8-7. On the other hand, if Alice and Bob play the game forever, Bob will win a majority of times. How is this even possible?*