# 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

P(male) =
110 / 200
P(male) =
102 / 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.

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