Algorithms to Live By

Laplaces Law tells us that we can actually make better guess of likelihood with a small n, by taking

o+1 / n+2 where o is the number of occurrences of some event, and n is the total number of attempts

plugging in real numbers aligns with our intuitions, if we win the lottery 3 times out of 6 plays we yield a 50% probability, but if we play 3 times and win 3 times, we yield a probability of 80%. This makes sense because, while we did win 100% of the games, we also understand that we didn’t play enough times to say the win rate is 100%. In fact, the only way to apply laplaces law and yield a 100% likelihood is by taking the limit as wins and plays go to infinity. In another situation imagine we win 0 times but play twice. The law yields 25%. As we take either variable to infinity (or to 0) we can see how our intuitions align.

an interesting boundary game is to imagine a new game I discover. I know nothing about its priors, I haven’t played it and I obviously haven’t won. The law immediately says we have a 50% chance of winning. The more we play the more we trust the odds