Conditional Probability

What is conditional probability?

\(P(B|A)\) is the probability of event B occuring given A has already occured.
Say you ahve a total of 25 friends. 10 of your friends like apples, 20 of your friends like oranges, while 5 of them like both apples and oranges. What is the probability that your friends while apples also like oranges?
Here let event A be friends liking apples, event B be friends liking oranges, then probability of friend who like apples, liking oranges will be \(P(B|A)\)

How to calculate conditional probability?

\(P(B|A) = \frac{P(A \cap B)}{P(A)}\)
Substituting the values in the above problem,

\(P(A \cap B) = \frac{5}{25}\, and\, P(A) = \frac{10}{25}\)

\(P(B|A) = \frac{\frac{5}{25}}{\frac{10}{25}} = \frac{1}{2}\)