# Joint probability

When you get the probability of the **intersection** of two events, we call this probability **joint probability**.

Let A and B be two events in a sample space.

The intersection of A and B is the collection of all outcomes that are common to both A and B.

We can denote the intersection of A and B as or AB

Take a look at the table on the left below.

Pass ={66, 54} and Males = { 66, 44 }

Now how do we compute probability of joint events?

Using the two tables below, we will compute P(pass / male)

In the lesson about probability of independent events, we found that

P(pass / male) = (left table) P(pass / male) = (right table)

There is another way to find these answers.

Notice that = (left table) = (right table)

Therefore, P(pass / male) = (left table) P(pass / male) = (right table)

P(pass and male) =

P(male) =

As you can see, it does not matter if the events are independent or not, the formula is

P(pass / male) =

**Multiplication rule** of joint events

Multiply both sides of P(pass / male) = by P(male)

P(male) × P(pass / male) = × P(male)

P(male) × P(pass / male) = P(pass and male)

P(pass and male ) = P(male) × P(pass / male)

## Joint probability of independent events

If A and B are independent events, we know that P(A) = P(A / B) or P(B) = P(B / A)

P(A and B) = P(A) × P(B /A).

Since P(B / A) = P(B), P(A and B) = P(A) × P(B)

## Joint probability of mutually events

The joint probability of two mutually exclusive events is always zero.

When two events A and B are mutually exclusive, = { }

In other words, the intersection is empty. Since the intersection is empty, the probability zero.