Law of Total Probability/Bayes' Theorem

[XodoX]

Can somebody explain to me, using an example, what those 2 theorems actually are? Like, when I see a problem, how do I know what I'm going to use?

I know Total Probability is "unconditional Probability", but I don't really get that.

Supose that F1, F2...Fn are events such that Fi$\bigcap$Fj=∅ whenever i≠j and F1$\bigcup$...$\bigcup$Fn=S. Then for any event E,

P(E)= P(E I F1) P(F1)+...+P(E I Fn) P(Fn).

Bayes' is for conditional probabilities, but apparently you calculate those conditional probabilities differently...

For any events E and F, the conditional probabilities P( E I F) and P(F I E) are connected by the following formula:

P(E I F)=P(F I E) P(E)/P(F)

The other definition of conditional probability was P(E I F)= P(E$\bigcap$F)/P(F). Can't figure out what the difference is, when I use which one..etc.

 

[kai_sikorski]

Supose that F1, F2...Fn are events such that Fi⋂Fj=∅ whenever i≠j and F1⋃...⋃Fn=S. Then for any event E,

P(E)= P(E I F1) P(F1)+...+P(E I Fn) P(Fn).

A lot of the time in probability problems it's easiest to break down the problem into mutually exclusive cases and deal with them separately. Like what's the probability that the sum of two dice is less than 6?

P(X₁ + X₂ ≤ 6) = P(X₂≤5)P(X₁=1) + P(X₂≤4)P(X₁=2) + P(X₂≤3)P(X₁=4) +P(X₂≤2)P(X₁=4) +P(X₂≤1)P(X₁=5)

So above in the sum you break down the cases based on the result of the first die.