What is the formula for calculating the probability of A given B?

The formula for calculating the probability of A given B is P(A|B) = P(A ∩ B) / P(B).

In probability theory, the notation P(A|B) represents the probability of event A occurring given that event B has already occurred. This is known as conditional probability. To calculate this, you need to know two things: the probability of both events A and B happening together, denoted as P(A ∩ B), and the probability of event B happening on its own, denoted as P(B).

The formula P(A|B) = P(A ∩ B) / P(B) essentially tells you how likely event A is, considering that event B has already taken place. It's important to note that P(B) must not be zero because you cannot condition on an event that has no chance of occurring.

For example, imagine you have a deck of 52 playing cards, and you want to find the probability of drawing an Ace (event A) given that you have drawn a red card (event B). First, you find P(A ∩ B), the probability of drawing a red Ace. There are 2 red Aces in the deck, so P(A ∩ B) = 2/52. Next, you find P(B), the probability of drawing any red card. There are 26 red cards, so P(B) = 26/52. Using the formula, P(A|B) = (2/52) / (26/52) = 2/26 = 1/13.

Understanding this formula helps you analyse situations where events are dependent on each other, which is a crucial concept in probability.