## Saturday, January 16, 2010

### Probabilistic Filler

Some important probabilities related to Scat:

As I mentioned last time, your chance of getting scat (no 1's) on your first roll is 3125/7776, or (5/6)5, which is just over 40% of the time. The probability, then, of having scat as your final hand is (5/6)15, or 6.5%, which works out to about twice every 31 turns.

The probability of rolling five 1's within three rolls is, in theory, about 1.3% (the exact probability simplifies to (91/216)5, which I think is kind of neat—not because that ratio has any significance, but because the otherwise-complicated calculation ends up simplifying so neatly), or about once every 75 turns. In practice, though, this probability is smaller, because of situations which dictate abandoning the all-1's strategy.

If there are k people still left to roll, then, the chance that at least one person gets scat is 1 - (1 - (5/6)15)k; and the maximum probability that at least one person rolls all 1's is 1 - (1 - (91/216)5)k. If, for example, four players were still to roll after you, there would be a 23.5% chance that at least one would roll scat (thus saving you from elimination, if you had a fairly low roll), and as much as a 5% chance that someone would end the game prematurely by rolling five 1's.

The main factor we must consider when deciding which dice to re-roll is the expectation of each decision; this expectation is specifically based on the probability of winning or losing the game after this decision. The probabilities above are part of calculating the expectation - you can make riskier moves (i.e., keeping only your 1's) when there are more players still to roll, because there is a higher chance that someone after you will get a bad hand, and because there is a higher chance that if you don't win right now, someone else will do so before you get another roll. I'll get more into the decision-making process next time.