Conditional Probability and Bayes Theorem

Conditional probability

The probability of an event B occurring when an event A has occurred is called conditional probability. It is denoted as P(B|A), and it is read as “The probability that B occurs given A has occurred”.

It is calculated using the formula below.

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P(B | A) = P (A $\cap$ B) / P(A) given P(A) > 0

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Here, P(A) is the probability of A and P(A $\cap$ B) is the joint probability of A and B.

Example

A math teacher gave her class two tests. 25% of the class passed both tests and 45% of the class passed the first test. What percent of those who passed the first test also passed the second test?

Solution

From the above question, we can deduce the following:

• Let A be the event that each class passed the first test. Then P(A) = 0.45

• Let B be the event that each class passed the second test.

• The joint probability that each class passed both tests is P(A $\cap$ B) = 0.25.

• P(B | A) = ?

P(B | A) = P (A $$ ...