Conditional probability helps us figure out how likely something is to happen based on another event that has already happened.

To find this probability, we use a simple formula:

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

Here,

• $P(A | B)$ means the chance of event A happening if we know event B has already happened.

Let’s look at some examples to make it clearer.

Example 1:

Imagine we have a deck of playing cards. We want to find out the chance of drawing an Ace, but we already know the card is a Spade.

1. There are 13 Spades in total.
2. Out of those, only 1 Ace is a Spade.

So, the probability of drawing an Ace given that it’s a Spade is:

$P(Ace | Spade) = \frac{1}{13}$

Example 2:

Now, let’s think about rain. If it rains (event B), what’s the chance that someone is carrying an umbrella (event A)?

• We know that 80% of people carry umbrellas when it rains.
• But on dry days, only 20% of people have umbrellas.

Here, understanding conditional probability helps us know how people usually behave in this situation.