How do we adjust our prior naive expectations?

[thegreatjared]

For example, if we had an arbitrary event that could either yield the result a or b, we might naively assign a probability of .5 to each result. After several trials, we can use the results of the trials to adjust our expectations of the probabilities of the event yielding a or b. If the first n trials result in b happening we expect b to happen in the next trial with probability (n+1)/(n+2).
My question is, how does this generalize to other such situations? If, let's say, there was another arbitrary event which had 5 possible outcomes, we might naively assign each result a probability of 1/5 prior to any trials. How would we then adjust our expectations as the trial results come in?

 

[micromass]

Search for Bayesian statistics.

 

[Stephen Tashi]

After several trials, we can use the results of the trials to adjust our expectations of the probabilities of the event yielding a or b. If the first n trials result in bhappening we expect b to happen in the next trial with probability (n+1)/(n+2).

That may be how you adjust your own expectations, but it isn't a theorem of probability theory. To justify your approach you need to assume more "givens" in the problem

As micromass suggests, look at the Bayesian approach to applying probability theory to problems. You must assume a "prior" distribution for the probabilities to justify any adjustment. The Bayesian approach can be applied to a situation with 5 outcomes. Look at the "maximum entropy" approach to selecting prior distributions.