Probability - Continuity
A Mathematical Perspective
Introduction to Probability
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.
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)! ]