Conditional Probability

In conditional probability, we allocate a distribution function to a sample space and then learn that an event E has occurred.
How should we modify the  probabilities of the remaining events? We shall call the new probability for an event F the conditional probability of F given E and denote it by P(F | E).

The conditional probability of event B occurs, given that event A has already occurred is

P(B|A) = P(A and B) / P(A)

Events of Conditional probability:

For example, we previously calculated the probability of rolling a 5 above. Now say we want to work out the probability of rolling a 5 given that one or both of the dice rolled is a 2. We would calculate this conditional probability like so

A = {(1, 4), (2, 3), (3, 2), (4, 1)}

B = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (1, 2), (3, 2), (4, 2), (5, 2), (6, 2)}

A AND B  =  {(2, 3), (3, 2)}

P(A | B)   =  P(A AND B) / P(B)

=  P({(2, 3), (3, 2)}) / P(B)

= (1/18) / (11 / 36)

=  (2/11)

Examples for conditional probability:

Example 1:

A math teacher gave her class two tests. 25% of the class passed both tests and 42% of the class passed the first test. What percent of those who passed the first test also passed he second test?

Solution:

P(Second | First)    

=        P(First and Second) / P(First)

=        0.25/0.42

=        0.60

=       60%

Example  2:

A jar contains black and white marbles. Two marbles are chosen without replacement. The probability of taking a black marble and then a white marble is 0.34, and the probability of taking a black marble on the first draw is 0.47. What is the probability of taking a white marble on the second draw, given that the first marble drawn was black?

Solution:

P(White | Black)          

=        P(Black and White) / P(Black)

=        0.34 / 0.47

=        0.72

=        72%