# Joint probability

The joint probability is the probability of the **intersection** of two events or the probability of two events happening together.

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 A ∩ B or AB.

The probability of A and B happening together is P(A ∩ B)

Take a look at the contingency table of independent events.

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

Pass ∩ Males = {66}

Now how do we compute the joint probability? Using the two tables below, we will compute P(pass / male)

Here is what we found In the lesson about probability of independent events.

Using the contingency table of independent events

P(pass / male) =

66
/
110

Using the contingency table of dependent events

P(pass / male) =

46
/
102

There is another way to find these answers.

Did you notice this using the table of independent events?

$$
\frac{\frac{66} {200}} {\frac{110} {200}} = \frac{66} {200} × \frac{200} {110} = \frac{66} {110} $$

Did you notice this using the table of dependent events?

$$
\frac{\frac{46} {200}} {\frac{102} {200}} = \frac{46} {200} × \frac{200} {102} = \frac{46} {102} $$

In conclusion,

$$ P(pass / male) =
\frac{\frac{66} {200}} {\frac{110} {200}} $$

In conclusion,

$$ P(pass / male) =
\frac{\frac{46} {200}} {\frac{102} {200}} $$

P(pass and male) =

66
/
200

P(pass and male) =

46
/
200

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

P(pass / male) =

P(pass and male)
/
P(male)

## Multiplication rule of joint events

Multiply both sides of the equation immediately above by P(male)

P(male) × P(pass / male) =

P(pass and male)
/
P(male)

× P(male)

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

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

In general, if A and B is the intersection of two events.

P(A and B) = P(A) × P(B / A) or P(A and B) = P(B) × P(A / B)

## 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 exclusive events

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

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

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