Conditional Probability

Conditional probability is the likelihood of an event A to occur given that another event that has a relation with event A has already occurred.

Suppose that we have 6 blue balls and 4 yellows placed in two boxes as seen below. I ask you to randomly pick a ball. The probability of getting a blue ball is 6 / 10 = 0,6. What if I ask you to pick a ball from box A? The probability of picking a blue ball clearly decreases. The condition here is to pick from box A which clearly changes the probability of the event (picking a blue ball). The probability of event A given that event B has occurred is denoted as p(A|B).

$$P(A|B) = \frac {P(A,B)}{P(B)} = \frac {P(A \cap B)}{P(B)}$$

Example

In a group of 100 sports car buyers, 40 bought alarm systems, 30 purchased bucket seats, and 20 purchased an alarm system and bucket seats. If a car buyer chosen at random bought an alarm system, what is the probability they also bought bucket seats?

Here, P(B) = Probability of a car buyer chosen at random bought an alarm system, which is 40/100 = 0.4 P(A,B) = Probability of both happening together, which is given 20/100 = 0.2 So, P(A|B) = 0.2/0.4 = 0.5