BAYES' THEOREM

                      BAYES' THEOREM

       Bayes’ Theorem is a way of finding a probability when we know certain other probabilities.

The formula is:

P(A|B) = P(A) P(B|A)P(B)

Which tells us:		how often A happens given that B happens, written P(A|B),
When we know:		how often B happens given that A happens, written P(B|A)
		and how likely A is on its own, written P(A)
		and how likely B is on its own, written P(B)

Let us say P(Fire) means how often there is fire, and P(Smoke) means how often we see smoke, then:

P(Fire|Smoke) means how often there is fire when we can see smoke
P(Smoke|Fire) means how often we can see smoke when there is fire

So the formula kind of tells us "forwards" P(Fire|Smoke) when we know "backwards" P(Smoke|Fire)

Example:

• dangerous fires are rare (1%)
• but smoke is fairly common (10%) due to barbecues,
• and 90% of dangerous fires make smoke

We can then discover the probability of dangerous Fire when there is Smoke:

P(Fire|Smoke) =P(Fire) P(Smoke|Fire)P(Smoke)

=1% x 90%10%

=9%