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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

P(A | B)

(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:

P(A | B) = P(A Ç B) / P(B)

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

Intersection of events

The probability of an intersection of events is calculated by the multiplicative rule, which makes use of conditional probabilities. We simply re-arrange the equation for the conditional probability P(B| A) = P(A Ç B) / P(A) and obtain:

P(A Ç B) = P(A) . P(B|A)


 

Example:

We have 10 marbles; 4 red and 6 blue, and take two of them randomly. We define the events A {the 1st marble is red} and B {the 2nd marble is red }. What is the probability that both marbles are red  P(A Ç B)?
Since we can take the marbles out one at a time, the probability of the 1st 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(AÇ 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:
 


probability tree

 

Last Update: 2006-Jšn-17