Dice odds seem simple at first glance, but I've never taken a Calculus based statistics course or game theory, and I think I may need to in order to solve some of the things I'm trying to solve. I can hammer out the odds in some of the more straight-forward scenarios, but when it comes to calculating the odds of a series of dice events with variable conditions... I get lost in the numbers. I've tried different, seemingly legit methods... only to return vastly different figures each time. Casino craps tables have a great variety of bets and hedge bets you can make. Some of the bets you make can sit on the table for half an hour or more, through dozens and dozens of dice throws, waiting on a resolution. It's technically possible (in theory not fact) that some of these bets could go on forever without being resolved. I would really like a decent approach to calculating these odds myself... hopefully by presenting a particular scenario, I can pick up enough tid-bits from your answers to piece together the methodology:
- We're throwing two six-sided dice at a time.
- Each side of each dice has equal probability on any given throw.
- There exists a tracking list with the numbers: 2, 3, 4, 5, 6, 8, 9, 10, 11, 12. (Note: 7 is not in this list.)
- When the dice are thrown and their total is a number other than 7, the number is crossed off the tracking list and the dice are rolled again.
- When the dice are thrown and their total is a number that's already been crossed off the tracking list, the dice are rolled again.
- At any point if the dice are thrown and their total is 7, the series is resolved as a loss.
- At any point if all ten numbers are crossed off the tracking list, the series is resolved as a win.
What's the probability of crossing all ten numbers off the list (winning) before throwing a 7 (losing)?
This is the"All or Nothing at All" bonus bet newly popular at many casinos. It pays 175 to 1. There are also "All Small" and "All Tall" bonus bets that pay 34 to 1 for throwing 2 thru 6 or 8 thru 12 respectively before throwing the 7. There's also a "Fire Bet" I'd like to break apart, but the rules are quite different. It will require a new post if I can't cull some new insights from the answers here... Please bear in mind, I'm wanting to know how to calculate conditional dice probability (where instantaneous probability shifts depending on your progression from throw to throw), not just know the odds in this particular case. I've taken mathematics courses up through CalcIII, so I can understand discussions involving limits and summation. Again, I've never takes statistics, probability, or game theory. Sorry for the long post, I know I talk too much....