Conditional Probability

A conditional probability is defined as the probability of an event, given that another event has occurred. This means that the probability for event A is effected by event B. Formally, a conditional probability is depicted as

(read: the probability of the event A under the condition that event B occurred).

Example:

We toss a die and define the events A {even number} and B {number is less than or equal to 3}. What is the probability of A if somebody gives us a hint that B has occurred? When B is true, we have the possible sample points 1, 2 and 3. So after given this information, the probability that the number is even is now 1/3. Without this prior information the probability would have been 1/2.

The calculation of the conditional probability P(A | B) involves two steps. First we have to realize that the fact that event B occurred reduces the sample space, to the sample points of B. This applies to the whole sample space which is now B and also the sample space of A. Only the sample points of event A that also belong to event B can occur thereafter; these are the sample points of the intersection of A and B. Since the probability is the ratio of the number of sample points of an event to the total number of sample points, the conditional probability is:

This is true under the condition that P(B) is not equal to zero. The equation adjusts the probability of A B from its original value in the whole sample space to the probability in the reduced sample space B.
 

Intersection of events

Example:

We have 10 marbles; 4 red and 6 blue, and take two of them randomly. We define the events A {the 1^st marble is red} and B {the 2^nd marble is red }. What is the probability that both marbles are red  P(A B)?
Since we can take the marbles out one at a time, the probability of the 1^st marble being red is 4/10. Getting two red marbles can then be seen as the conditional probability of getting a second red marble P(B|A), given the first one is red. After the removal of the first marble, the sample space has changed: we now have 3 red and 6 blue, so the probability of getting a red one now is P(B|A) = 3/9.
P(A B) = P(A) . P(B|A) = 4/10 * 3/9 = 2/15. The calculation of the probability of intersections can be displayed in a tree diagram: