# Dependent Events

How to find the probability of dependent events: formula, example, and its solution.

## Formula

Dependent events are the events

that do affect each other.

If *A* and *B* are dependent events,

then P(*A* and *B*) can be found

by multiplying P(*A*) and P(*B*'):

P(*A* and *B*) = P(*A*)⋅P(*B*').

P(*A* and *B*): Probability of both *A* and *B* happening

P(*A*): Probability of *A* happening

P(*B*'): Probability of *B* happening affected by *A*

You can see that

P(*B*') part is different

from the independent events formula.

So you should keep attention

when finding P(*B*').

Independent events

## Example

After the first pick,

the picked number is not replaced.

So the first pick do affects the second pick.

So these two events are dependent events.

So find P(*A*),

find P(*B*'), the probability affected by *A*,

then find P(*A* and *B*)

by multiplying the probabilities.

First see the first pick.

Numbers from 1 to 10 are given.

So there are 10 numbers

that can be picked.

So n(*S*) = 10.

Set the event *A* as

picking the multiples of 3 at the first pick.

The multiples of 3 from 1 to 10 are

{3, 6, 9}.

So n(*A*) = 3.

n(*S*) = 10

n(*A*) = 3

So P(*A*) = 3/10.

Probability

Next, see the second pick.

The picked number is not replaced.

So there are, 10 - 1, 9 numbers

that can be picked.

So n(*S*'), the changed space sample, is 9.

Set the event *B* as

picking the multiples of 3 at the second pick.

One of the multiples of 3 is already picked

and not replaced.

So the remaning multiples of 3 are

{6, 9} or {3, 9} or {3, 6}.

So n(*B*') = 2.

n(*S*') = 9

n(*B*') = 2

So P(*B*') = 2/9.

It's good to compare this

with the independent events problem.

Because of no replacement,

P(*B*') part has changed.

Independent events - Example 3

P(*A*) = 3/10

P(*B*') = 2/9

So P(*A* and *B*) = (3/10)⋅(2/9).

Cancel 3 and reduce 9 to 3.

Cancel 2 and reduce 10 to 5.

Then (3/10)⋅(2/9) = (1/5)⋅(1/3).

So 1/15 is the answer.