# Probability is Magical

## What is probability?

It's a measure or estimation of how likely it is that something will happen or that a statement is true. Probabilities are given a value between 0 (0% chance or will not happen) and 1 (100% chance or will happen). The higher the degree of probability, the more likely the event is to happen, or, in a longer series of samples, the greater the number of times such event is expected to happen.

## Example: Throwing the dice

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

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

if we want to check the chances of getting a "4"

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 1/6

## What is Counting

An efficient way of counting is necessary to handle large masses of statistical data, and for an understanding of probability.

Let the number of possible outcomes of an event be n(E).

If the two possible events E1 and E2 have no common outcomes then n(E)=n(E1)+n(E2)

If event E1 can result in any one of n(E1) possible outcomes, and for each outcome of the event E1, there are n(E2) possible outcomes of E2 then n(E)=n(E1) x n(E2)

## What is Factorial Notation?

Factorial notation is used to write the product of all the positive whole numbers up to a given number.

n factorial is the product of all the integers from 1 to n

"n factorial" is written with an exclamation mark as follows: n!

n! = (n)(n − 1)(n − 2)...

## Example:

6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

Counting & Permutations - Probability 1

## What is permutation?

A permutation is an arrangement of objects, without repetition, and order being important. Another definition of permutation is the number of such arrangements that are possible.

Since a permutation is the number of ways you can arrange objects, it will always be a whole number. The denominator in the formula will always divide evenly into the numerator.

The n value is the total number of objects to chose from. The r is the number of objects your actually using.

The two key things to notice about permutations are that there is no repetition of objects allowed and that order is important.

## Example: Permutations of the letters ABCD

ABCD
ABDC
ACBD
ACDB

BACD
BCDA
BDAC
BDCA

CABD
CBDA
CDAB
CDBA

DABC
DACB
DBAC
DBCA
DCAB
DCBA

Now, if you didn't actually need a listing of all the permutations, you could use the formula for the number of permutations. There are 4 objects and you're taking 4 at a time. 4P4 = 4! / (4-4)! = 4! / 0! = 24 / 1 = 24.

This also gives us another definition of permutations. A permutation when you include all n objects is n!. That is, P(n,n) = n!

## What are Combinations?

A combination is an arrangement of objects, without repetition, and order not being important. Another definition of combination is the number of such arrangements that are possible.

nCr = n! ÷ [r! (n - r)!]

The n and r in the formula stand for the total number of objects to choose from and the number of objects in the arrangement, respectively.

The key points to a combination are that there is no repetition of objects allowed and the order isn't important.

## Example:

What is the number of different sets of 4 letters which can be chosen from the alphabet?

There are 26 letters in the alphabet:

26 C 4 = (26!) / [4! (26-4)!] = 14950