A Peek into Probability

A Pre-lesson look at Probability and Counting

Why would mathmaticians study the concept of probability ?

"Most likely" "probably" and "a poor chance" are only some of the terms used in daily life all referring to probability. With the study of probability, these terms are used, more carefully and accurately and thanks to our friends the mathematicians we can now really know our chances when doing a bet or taking a risk.

Multiplication

Consider the case when two occurrences E1, E2 are to be performed and that they are independent occurrences (one outcome doesn't affect the other's outcome)


(An example is the jeans, t-shirts and heels idea shown in the video E1 being jeans, E2 t-shirts, and E3 heels)



General rule:

Suppose that occurrence E1 can result in any one of n(E1) possible outcomes; and for each outcome of the occurrence E1, there are n(E2) possible outcomes of occurrence E2. So if we assign E as the occurrence that both E1 and E2 must occur, then


n(E) = n(E1) × n(E2)

Multiplication shows us the many possibilities there are when dealing with 2 or more different occurrences.



Factorial Notation:

The factorial notation represents the product of all the consecutive numbers noted as n, (n-1), (n-2)... with n being the greatest number; up to were the product is taken and such that all these numbers belong to the set of natural numbers (strictly positive whole number). The notation is as such:



n! = (n)(n − 1)…(1)



NOTE: YOU CAN’T SIMPLIFY A FRACTION OF FACTORIALS. ex: ( 4!/ 2! 2!)

Ex. 4!/ 2! = 24/ 2 =12


Permutations with identical elements:

Concept A)

In general, n distinct elements can be arranged in n! ways.


Concept B)

The number of permutations of n distinct elements in E taken r elements of E at a time, denoted by nPr, without repetitions, is represented as:


nPr=n(n−1)(n−2)...(nr+1)=n!/(nr)!


EX. How many words can be formed from the letters of my first name "LANA"?

4! / (4-2)! = 4P2= 12


Concept C)

The number of different permutations of n elements of which n1 and n2 are two different kinds of elements, with nk being the k-th kind of element, is found as such:

n!n1!×n2!×n3×...×nk!


Concept D)

The number of permutations where repetition is considered of a set E consisting of n elements is equal to n!

Permutations