conditional probability

. Conditional probability

Many times we know additional information that affects the calculation of a probability:
• What is the probability that a person voted for Obama if you know he is a registered Republican?
• What is that probability that the a person is guilty of murder if you know the crime was committed by a left-handed person? If the person is right handed, the probability is 0. If the person is left handed, is the probability is 1?

The probability that A occurs given that B occurs is called the conditional probability of A given B and is

written P(A|B) as
P(A|B) = (P(A?B))/(P(B)); for P(B) > 0:
Note that we can only condition on events with positive probability.
Under the classical interpretation of probability, we can see that if we are told that B has occurred, then all outcomes in B are equally likely, and all outcomes not in B have zero probability

Example: A fair coin is tossed two times. What is the probability that the second coin is a head if you know that at least one head appears.
The four outcomes are:
H H , H T , T H
T T
We have checked those outcomes which satisfy the condition that at least one head occurs.
Note that the fourth outcome (TT) in the table is not checked. Out of the three events that satisfy this condition (HH, HT, TH) exactly two have heads for the second toss
P(second toss is a head given at least one head) = 2/ 3