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Figure 1. We cannot always say much about a distribution since they are not always easily parameterized, but Tchebysheff's Theorem states neatly that the area under a density function within k standard deviations of the mean is 1 - 1/k^2.
Figure 2. Bayes' Theorem is basically this. The conditional probability - i.e. the probability that the observation x occurs assuming that event A takes place - is the classic probability of x occurring if the sample space were A rather than X. In the formula, the division by P(A) is to scale probabilities that concur with A so that P(x|A) is a new probability distribution.
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