# Probability

How to find the probability of an event: definition, examples, and their solutions.

## Definition

The probability of an event can be found

by using the formula below.

P(*A*) = n(*A*)/n(*S*)

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

n(*A*): Number of ways of the event *A* happening

n(*S*): Total number of ways

(= Number of space sample)*S* stands for the [sample space].

The probability has a value between 0 and 1.

P(*A*) = 0 means

the event *A* doesn't happen.

P(*A*) = 1 means

the event *A* always happens.

So 0 ≤ P(*A*) ≤ 1.

## Example 1

A die has 6 sides: from 1 to 6.

So there are 6 ways to get a number.

So n(*S*) = 6.

Set the event *A* as getting a 3.

There's 1 way to get a 3.

So n(*A*) = 1.

n(*S*) = 6

n(*A*) = 1

So the probability of getting a 3 is

P(*A*) = 1/6.

## Example 2

A die has 6 sides: from 1 to 6.

So there are 6 ways to get a number.

So n(*S*) = 6.

Set the event *A* as

getting an odd number.

The odd numbers from 1 to 6 are

{1, 3, 5}.

There are 3 ways

to get an odd number.

So n(*A*) = 3.

n(*S*) = 6

n(*A*) = 3

So the probability of getting an odd number is

P(*A*) = 3/6.

Divide the numerator and the denominator by 3.

Then 3/6 = 1/2.

## Example 3

Numbers from 1 to 30 are given.

So there are 30 numbers

that can be picked.

So n(*S*) = 30.

Set the event *A* as

picking a prime number.

The prime numbers from 1 and 30 are

{2, 3, 5, 7, 11, 13, 17, 19, 23, 29}.

There are 10 prime numbers

that can be picked.

So n(*A*) = 10.

n(*S*) = 30

n(*A*) = 10

So P(*A*) = 10/30.

Divide the numerator and the denominator by 10.

Then 10/30 = 1/3.

## Example 4

The spinner is divided into five parts:

from 1 to 5.

And each part has the same area.

So there are 5 ways

that the arrow can land.

So n(*S*) = 5.

[*X* = 2] means

the event of landing at 2.

There's only 1 way

that the arrow can land at 2.

So n(*A*) = 1.

n(*S*) = 5

n(*X* = 2) = 1

So P(*X* = 2) = 1/5.

## Example 5

The pizza is divided into 8 pieces.

So n(*S*) = 8.

Set the event *A* as

choosing the piece with pepperoni.

There are 5 pieces

that have pepperoni on them.

So n(*A*) = 5.

n(*S*) = 8

n(*A*) = 5

So P(*A*) = 5/8.