Conditional Probability: Explained with a simple example

The easiest way to understand conditional probability is to start with the definition and to look at an example.

Definition: Let E and F be two dependent random variables or events. Probability of E occurring given that F has occurred [denoted by P(E|F)] is called the conditional probability of E given F.

P(E ∩ F) = P(E |F)*P(F) .

Example: Let’s consider a scenario where there are 20 cards numbered 1 to 20 in a hat. A card is drawn at random from the hat. We are interested in the conditional probability of the card drawn being greater than 15, if we are told that the card drawn is greater than 10.

Solution: Let E and F denote the events that the card drawn is greater than 15 and 10, respectively.

Note that we are interested in determining P(E|F), which is the conditional probability of the card drawn being greater than 15, if one is told that the card drawn is greater than 10.

Now, P(E ∩ F) = P(E) = (20–15)/20 = 5/20, because it denotes the probability that is card drawn is above 15 as well as above 10. As F denotes the random variable that the number drawn is greater than 10,

P(F) = (20–10)/20 = 10/20.

Therefore,

P(E|F) = P(E ∩ F)/ P(F) =(5/20)/(10/20) = 5/10 = 1/2, which is the value that we are interested in.

One can learn more about conditional probability in this video.