**Theory of Conditional Probability**

This is based on the concept of Dependent events, where the probability that an event B takes place will depend on whether another event A has or has not taken place

The conditional probability of an event B, given the event A; denoted by P(B/A) is defined as

P(B/A) = P(A ∩ B)/P(A) where P(A) ≠ 0

P(A ∩ B) is the joint probability of A and B, or the probability of both A and B together.

Let us understand this formula with an example

**E.g 9) 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 the second test? **

This problem describes a conditional probability since it asks us to find the probability that the second test was passed given that the first test was passed. P(Second/First) = P(First ∩ Second)/P(first) = 0.25/0.42 = 0.60 = 60%

**Illustrations **

**E.g. 10) The probability that it is Thursday and that a student is absent is 0.03. There are 5 school days in a week. What is the probability that a student is absent given that today is Thursday? **

Since there are 5 school days in a week, the probability that it is Thursday is 0.2

P(Absent/Thursday) = P(Absent ∩ Thursday)/P(Thursday) = 0.03/0.2 = 0.15 = 15% E.g.

**11) At a CBSE school, 18% of all students play football and basketball and 32% of all students play football. What is the probability that a student plays basketball given that the student plays football?**

B → Basketball F → Football

P(B/F) = P(B ∩ F)/P(F) = 18/32 = 0.56 = 56%

**CHECKPOST **

6. In Del dorado, 84% of the houses have a garage and 65% of the houses have a garage and a back yard. What is the probability that a house has a backyard given that it has a garage?

7. 56% of all children get an allowance and 41% of all children get an allowance and do gardening. What is the probability that a child does gardening given that the child gets an allowance?

8. The probability that a student takes Technology and Spanish is 0.087. The probability that a student takes Technology is 0.68. What is the probability that a student takes Spanish given that the student is taking Technology?