|
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) | |||||
|
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 : |
|
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 Madonna to Josh? |
||||||||
|
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 :
|
|