I have a (I suspect unusual) tendency to look at basic concepts and try to see them in as many ways as possible. For example, here are seven equations, all of which could be referred to as Bayes' Theorem:

=\frac{P(E%7CH).P(H)}{P(E)}\\[10]P(H%7CE)=\frac{P(E%7CH)}{P(E)}.P(H)\\[10]P(H%7CE)=\frac{P(E%7CH).P(H)}{P(E%7CH).P(H)+P(E%7C\neg%20\!\,H).P(\neg%20\!\,H)}\\[10]P(H%7CE)=\frac{1}{1+\frac{P(E%7C\neg%20\!\,H).P(\neg%20\!\,H)}{P(E%7CH).P(H)}}\\[10]P(H%7CE)=\frac{P(E%7CH).P(H)}{\sum%20P(E%7CH_i).P(H_i)}\\[10]%0Aodds(H%7CE)=\frac{P(E%7CH)}{P(E%7C\neg%20\!\,H)}.odds(H)\\[10]%0Alogodds(H%7CE)=log(\frac{P(E%7CH)}{P(E%7C\neg%20\!\,H)})+logodds(H))

However, each one is different, and forces a different intuitive understanding of Bayes' Theorem. The fourth one down is my favourite, as it makes obvious that the update depends only on the ratio of likelihoods. It also gives us our motivation for taking odds, since this clears up the 1/(1+x)ness of the equation.

Because of this way of understanding things, I find explanations easy, because if one method isn't working, another one will.

ETA: I'd love to see more versions of Bayes' Theorem, if anyone has any more to post.