Bayes Rule

Given Events E1 and F, write two conditional probabilities :

P(E1 | F) = P(E1 F)

P(F)
or P(E1 F) = P(E1 | F)P(F)
P(F | E1) = P(E1 F)

P(E1)
or P(E1 F) = P(F | E1)P(E1)

 left sidesequal above
means
 right sidesequal below

P(E1 | F)P(F) = P(F | E1)P(E1)

Divide both sides of the above by P(F) , to get a SIMPLE FORM OF BAYES RULE :
 P(E1 | F) = P(E1)P(F | E1) P(F)

The denominator P(F) above can be re-written if the sample space (and hence F) are partitioned, or divided, into 2 or more non-overlapping pieces, as implied in the classic box-cutting magic trick, and its representation using sets in the figures (right) :
Don't worry : This won't hurt at all
Prefer Josh to Madonna?
 E1 E2 E3

If E1 , E2 , and E3 do not overlap, then (F E1) , (F E2) , and (F E3) also do not overlap, and hence
P(F) = P(F E1) + P(F E2) + P(F E3) ; re-writing each term as a product (as at the top of this page) :

P(F) = P(E1)P(F | E1)   +   P(E2)P(F | E2)   +   P(E3)P(F | E3) ;

We can then re-write the simpler form (above) to a LARGER FORM OF BAYES RULE :

 P(E1 | F) = P(E1)P(F | E1) P(E1)P(F | E1) + P(E2)P(F | E2) + P(E3)P(F | E3)

 This page lastupdated 7 October 2013