Bayes' Theorem?

Question

Could anyone explain to me and prove to me Bayes' Theorem?

Answer 1

It's rather long to go through but I imagine you'll find this to be helpful:

http://en.wikipedia.org/wiki/Bayes'_theorem

Answer 2

Draw 2 overlapping circles, Venn diagram style, labelled A and B. In the overlap, put 30. In the part of A that isn't B, put 10. In the part of B that isn't A, put 30. (all percents, if you like, probabilities).

so P(A|B) = P(AB)/P(B) = 30/60 = 1/2
and P(B|A) = P(AB)/P(A) = 30/40 = 3/4

That puts the basic arithmetic into a clear picture. From the 2 equations,

P(A|B) P(B) = P(AB) = P(B|A) P(A)
P(A|B) = P(B|A) P(A) / P(B)

and that's Bayes' Theorem. But do see the link.

Answer 3

Bayes' theorem describes the relationships that exist within an array of simple and conditional probabilities. For example: Suppose there is a certain disease randomly found in one-half of one percent (.005) of the general population. A certain clinical blood test is 99 percent (.99) effective in detecting the presence of this disease; that is, it will yield an accurate positive result in 99 percent of the cases where the disease is actually present. But it also yields false-positive results in 5 percent (.05) of the cases where the disease is not present.