When you get the chance of **overlap** of two events we call this probability **common probability**.

Let A and B be two events in an example room.

The intersection of A and B is the collection of all the results that A and B have in common.

We can use the intersection of A and B as A. describe ∩ B or AB.

Take a look at the contingency table of independent events.

Passport = {66, 54} and men = {66, 44}

Consist ∩ Men = {66}

So how do we calculate the probability of common events?

With the following two tables we calculate P (pass / male)

Here’s what we found in the lesson on the likelihood of independent events.

Using the contingency table of independent events

P (passport / male) =

66

110

Use the contingency table of dependent events

P (passport / male) =

46

102

There is another way to find these answers.

Did you notice this from 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 from the table of dependent events?

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

Finally,

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

Finally,

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

P (passport and male) =

66

200

P (passport and male) =

46

200

As you can see, it doesn’t matter whether the events are independent or not, the formula is

P (passport / male) =

P (passport and male)

P (male)

## Multiplication rule of common events

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

P (male) × P (passed / male) =

P (passport and male)

P (male)

× P (male)

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

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

In general, when A and B are the intersection of two events.

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

## Common 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)

## Common probability of mutual events

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

If two events A and B are mutually exclusive, then A holds ∩ B = {}

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