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:
Big image

So the general formula:

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

Math Made Easy: Probability

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)! ]

Permutations