# Probability of Independent Events: Definition, Formula, Examples

## What Is Independent Event?

Independent means not relate with something/someone. For examples of independent condition are buying a chocolate and finding a pencil on the road, writing a letter and sweeping the floor, etc. The events in examples of independent condition are not relate each other. It is not causal relationship between of them.

## Probability of Independent Events Formula

Probability is chance to happen of something/events. Probability equals with number of something happen divide by total numbers of outcomes. Probability of independent event means probability that is not related or affected with previous or next events. Some examples of independent events are: tossing coins, rolling dices, deck of cards, and combined between them.

If there are two independent events, A and B, then the first step is determining the probability of each events. Then the probability of independent evets is

P(AB) = P(A) x P(B)

Probability of independent events of A and B is gotten by multiplying probability of A and B.

How if there are more than two events?

Basically, the concept is same with two events. It is just multiplying the probability of each events.

## Examples

1. There is a dice that is rolling twice. Find the probability of getting the sum is 6.

The events that shows the sum is 6:

(1,5), (2,4), (3,3), (4,2), and (5,1)

There are 5 events.

The total numbers of outcomes based on the problem is 6 x 6 = 62 = 36.

Then the probability of independent events is 5/36.

2. A coin is tossed four times. Find the probability of getting two tail and two heads.

There is a coin that is tossed four times, so there are 24 = 16 total numbers of outcomes.

Total events of two tail and two heads are:

HHTT, HTHT, TTHH, THTH, HTTH, THHT

There are 6 events

Then, the probability is 6/16 = 3/8

3. A code consists of a digit number chosen from 0 to 9 and followed by two different letters of the alphabet.
What is the probability the code is 2AB?

• Total number of digit number is 10 (from 0 to 9).

Then P(2) = 1/10

• Total number of letters are 26.

P(A) = 1/26

P(B) = 1/25

(it is because of the letter is different, so the second letter is gotten from 25 letters not 26 letters anymore)

Then P(AB) = 1/26 x 1/25 = 1/650

• So, the probability is

P(2AB) = 1/10 x 1/650 = 1/6500

Another way is

P(2∩A∩B) = P(2) x P(A) x P(B) = 1/10 x 1/26 x 1/25 = 1/6500.

4. A jar contains 4 black, 3 yellow, 4 white and 1 purple marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. What is the probability of choosing a black and then two white marbles?

Number of each marble:

Black: 4

Yellow: 3

White: 4

Purple: 1

Total marble is 12.

Probability of black marble: ¼

Probability of two white marbles: 2/4 = ½

Then, the probability of a black and two white marbles:

P(B∩2W) = ¼ x ½ = 1/8