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Conditional Probability Formula

Conditional Probability is a measure of the probability of an event given that another event has occurred.

If the event of interest is A and the event B is known or assumed to have occurred, “the conditional probability of A given B”, or “the probability of A under the condition B”, is usually written as P(A|B), or sometimes PB(A).

When B is given by A, then conditional probability is,

\[\LARGE P(B|A)= \frac{P(A\cap B)}{P(A)}\]

When A is given by B, then the conditional probability is

\[\LARGE P(A|B)= \frac{P(A\cap B)}{P(B)}\]

Solved Examples

Question 1:
The probability that it is Friday and that a student is absent is 0.03. Since there are 5 school days in a week, the probability that it is Friday is 0.2. What is the probability that a student is absent given that today is Friday?

Solution:
Formula of Conditional probability Formula

$P(B|A)= \frac{P(A\cap B)}{P(A)}$
$P(Absent|Friday)= \frac{P(Absent\: and \: Friday)}{P(Friday)}$
= $\frac{0.03}{0.2}$
= 0.15
= 15 %