# Probability - Continuity

### A Mathematical Perspective

## Introduction to Probability

**Probability**is the chance of something happening. The likelihood or

**chance**that a certain event will happen or not. It is the ratio of the

**number of ways an event can occur**to the

**number of possible outcomes.**

## Let's take a simple example:

When a dice is thrown, there are six possible outcomes: **1, 2, 3, 4, 5, 6**.

The probability of any one of them is 1/6.

**If you flip a coin:**

## So the general formula:

Probability of an event happening = Number of ways it can happen\Total number of outcomes

## 1. Factorial notation

Factorial notation is the product of all positive integers less than or equal to n. the factorial of a non-negative integer n, denoted by n!: *n*! = (*n*)(*n* − 1)(*n* − 2)...(3)(2)(1)

For example,

7! = 7 × 6 × 5 × 4 × 3 x 2 x 1 = 120

Note: By convention 0! = 1

Also, we cannot simply cancel a fraction containing factorials: 10!\5! is not equal to 2!

## 2. Permutation - Order does matter here!

Informally, a permutation of a set of objects is an arrangement of those objects into a particular order, or all the **possible** ways of doing something.

A- The number of permutations of *n* distinct objects is *n*×(*n* − 1)×(*n* − 2)×⋯×2×1, which is commonly denoted as "*n* factoriall" and written "*n*!".

B-The number of permutations of n distinct objects at a certain: r. (denoted by nPr)

**nPr = n!/(n-r)**

**Example: **

How many ways can first and second place be awarded to 10 people?

**10!**=**10!**=**3,628,800**

**= 90**

**(10-2)!****8!****40,320**

How many ways can first and second place be awarded to 10 people?

**10!**=**10!**=**3,628,800**

**= 90**

**(10-2)!****8!****40,320**

## Combinations - Order Does matter here!

Combination of n objects taken at a certain: r.

I repeat: order does NOT matter here. However repition is not allowed.

Number of combinations: nCr = n! [ r! (n-r)! ]