Conditional Probability

Conditional probabilities are the probabilities that one event happens given that another event has already happened.

It’s denoted as P(A | B) and read as “the probability of event A given [Event] B [has happened].” It’s calculated by finding the probability that both Events A and B happen (P(A ∩ B)) divided by the probability of just Event B happening (P(B)).

From past experience, we know there’s a 72% chance we’ll both eat too many corns dogs and get an upset stomach when Skeevy’s Carnival comes to town. We also know that that there’s a 95% chance we’ll eat too many corn dogs. We want to know the probability that we'll get an upset stomach due to eating too many corn dogs.

We divide the probability of both events happening (0.72) by the probability that we eat too many corn dogs (0.95). 0.72 ÷ 0.95 = 0.758. There’s a 75.8% chance we’ll get an upset stomach from eating too many corn dogs. If we want that probability to be 100%, however, we just need to take a few rides on the Gravitron.

Another example: Let’s say you have an apple, a banana, and an orange in a bag. The probability of pulling out any of them is 1 in 3, or 33%. But if you first pull out an apple, the probability of picking a banana or orange next goes up to 50%, or one out of two. So the chance of drawing that orange after the apple is drawn is 33% x 50% = 16.5%.

Or, in more real life terms, Larry is applying to be a law clerk at a prestigious law firm. Out of 2,000 applicants they accept 200 clerks, or 10%. But only 2% are paid clerkships. Half of these (50%) also receive a travel and clothing allowance. The chance of Larry being hired and receiving a salary is .2% (.1 x .02). The chance of Larry being hired, receiving a salary, and then also receiving a travel and clothing allowance is .1% (.1 x .02 x .5). Perhaps Larry should keep looking for a better offer.