# Probability: Dependent Events

How to find the probability of dependent events: formula, 1 example, and its solution.

## Formula

### Formula

Independent events are the events

that do affect each other.

If A and B are dependent events,

then

P(A and B) = P(A)⋅P(B').

P(A): Probability of A happening

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

If A is also affected by B,

then change P(A) to P(A').

## Example

### Example

### Solution

First, let's see the 1st pick.

There are 5, 4, and 3 marbles in the jar.

So n(S) = 5 + 4 + 3.

5 + 4 = 9

9 + 3 = 12

So n(S) = 12.

This means,

for the 1st pick,

there are 12 marbles.

There are 5 blue marbles.

So n(A) = 5.

n(S) = 12

So P(A) = 5/12.

Probability

Next, let's see the 2nd pick.

It says

a marble is randomly picked from the jar

and [not replaced].

So the 1st pick (A)

does affect the 2nd pick (B).

So A and B are dependent events.

So, instead of finding P(B) by

P(B) = P(A) = 5/12,

find the changed probability P(B').

There were 5 blue marbles.

And 1 blue marble is picked and not replaced.

Then there are, 5 - 1, 4 blue marbles.

So n(B') = 4.

There were 12 blue marbles. (n(S) = 12)

And 1 marble is picked and not replaced.

Then there are, 12 - 1, 11 marbles.

So n(S') = 11.

n(B') = 4

n(S') = 11

So P(B') = 4/11.

P(A) = 5/12

P(B') = 4/11

A and B are dependent events.

So

P(A and B) = [5/12]⋅[4/11].

Cancel the numerator 4

and reduce the denominator 12 to, 12/4, 3.

3⋅11 = 33

So 5/33 is the answer.