Conditional Probability
More often than not, we wish to express probabilities
conditionally. i.e. we specify the assumptions or
conditions under which it was measured
Conditional statements of probability take the form:
Pr{H|O} ≡ probability that event
H occurred in the subset of
cases where event O also
occurred
Conditional Probability
This can be represented as a Venn diagram
S
H
O
The intersection H ∩ O represents events
where both H and O occurred
Conditional Probability
More often than not, we wish to express probabilities
conditionally. i.e. we specify the assumptions or
conditions under which it was measured
We can express the idea shown in the Venn diagram as:
P(H | O ) = P(H & O) / P(H)
Note that in the “universe of possibilities”, O has effectively
replaced S. Our probability for event H is now conditional
on the assumption that event O has already taken place.
Tree Diagram Representations
H&O
O
P(H & O) = P(H|O) P(O)
~H & O
S
H & ~O
~O
~H & ~O
See www.mathsisfun.com/data/probability-events-conditional.html
Bayes’ Theorem
Suppose we observe H but do not observe whether O occurred or not.
In order to infer O we may use the previous tree diagram
H&O
O
~H & O
S
H & ~O
~O
~H & ~O
P(O|H)
= P(H & O) / (P(H & O) + P(H & ~O))
= P(H|O)P(O) / (P(H|O)P(O) + P(H|~O)P(~O))
See www.digitalbiologist.com/2014/01/a-theorem-for-all-seasons.html