CONDITIONAL PROBABILITY

Definition Of Conditional Probability

The Conditional Probability of an event A, assuming that B has already occurred is given by P(A|B) = .

More About Conditional Probability

P(A|B) is read as "the probability of A, given B".
If the probability of an event that has already occurred is 0, then P(A|B) is undefined.

Video Examples: Math: Conditional Probability

Example of Conditional Probability

There are 6 red marbles, 5 green marbles, and 4 yellow marbles in a bag. Suppose Ed picks a red marble first without replacement. Again, he wants to pick a red marble next. Since the first marble selected is not replaced back, the total number of marbles is reduced by one. Now there are 5 red marbles in the bag of 14 marbles. So the Conditional Probability P(picking a red marble next) = = 5/14.

Solved Example on Conditional Probability

Ques: A box contains black balls and white balls. A person selects two balls without replacement. If the probability of selecting a black ball and a white ball is 15/56 , then find the probability of selecting a white ball on the second draw, given that the probability of selecting a black ball on the first draw is 3/8.

Choices:

A.5/7
B.7/5
C.1/8
D. 45/448
Correct Answer: A

Solution:

Step 1: Let the task of selecting black ball be B.
Step 2: Let the task of selecting white ball be W.
Step 3: Given probability of selecting a black ball on the first draw P(B) =3/8
Step 4: Probability of selecting a black ball and a white ball P(B and W) =15/56
Step 5: Probability of selecting white ball on the second draw with black ball on the first draw P(W|B)=  [P(B|A) = .]
Step 6: P(W|B) =
Step 7: P(W|B) =15/56 × 8/3 =5/7
Step 8: The probability of selecting a white ball on the second draw, given that the first ball selected was a black ball is 5/7.