[Rich &Knight, ch 8]
In the logic based approaches above we have still assumed that everything is either believed false or believed true. However, it is often useful to represent the fact that we believe that something is probably true, or true with probability (say) 0.65. This is useful both for dealing with problems where there is genuine randomness and unpredictability in the world (such as in games of chance) and also for dealing with problems where we could, if we had sufficient information, work out exactly what is true in the world, but where this is impractical. As an example of the latter type of problem, in our elephant example we might choose, rather than listing all non-grey elephants, just to have a rule which concludes that elephants are grey with probability 0.9 (90%). The fact that Clyde is an elephant would be treated as evidence suggesting that he is likely to be grey, which might be balanced with competing evidence to determine the total likelihood of greyness.
To do all this in a principled way requires techniques for probabalistic reasoning. In this section we'll briefly introduce classical Bayesian probability theory, then discuss how uncertainty was (and still is) treated in many expert systems. We'll conclude by briefly mentioning other important approaches to uncertainty.