# Counting and Probability

### Do you think counting is just 1,2 to ∞?What is probability?

## Introduction

Everyday you count whether you are seeing how much money you have or how many subjects you have in a particular day … Also everyday you hear words like probably, more likely, less likely, chance. These are all related to probability.

Probability is when we are not sure something will happen. It is the comparison between the number of times an event could occur and the total number of possible events. While counting is naming or listing the units of a group or collection one by one in order to determine a total.

## What is n!? What are the basic principles of counting?What are permutations?

1) **n****factorial** (n!) is defined as the product of all the integers from 1 to *n.*

* n*! = (*n*)(*n* − 1)(*n* − 2)...(3)(2)(1)

ex: 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

2) **Basic Principles of Counting:**

-Addition Rule: *E* = "day of the week"

*n*(*E*)= "number of outcomes of event *E*"

Let *E*1 and *E*2 be mutually exclusive (there are no common outcomes)** **events.

The number of times event *E* will occur is: *n*(*E*) = *n*(*E*1) + *n*(*E*2)

- Multiplication Rule:

Let *E*1 and *E*2 be two** **independent (one does not affect the other's outcome) events.

There will be *n*(*E*1) × *n*(*E*2) possible outcomes of the two events. *n*(*E*) = *n*(*E*1) × *n*(*E*2)

3) **Permutations: **

A permutation is an arrangement (or ordering) of a set of objects. There are 4

theorems:

-Theorem 1 - Arranging *n* Objects

-Theorem 2 - Number of Permutations

-Theorem 3 - Permutations of Different Kinds of Objects

-Theorem 4 - Arranging Objects in a Circle

For more detailed information, please visit http://www.intmath.com/counting-probability/counting-probability-intro.php

## Some Examples

## Addition RuleIn how many ways can a number be chosen from 1 to 31 such that it is a multiple of 3 or 9? Let E1 = multiples of 3 E1 = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30} n(E1) = 10 E2 = multiples of 9: E2 = {9, 18, 27} n(E2) = 3 Events E1 and E2 are mutually exclusive (there are no common outcomes). n(E) = n(E1) + n(E2) = 10+3 = 13 There are thirteen ways for a number to be chosen from 1 to 31 such that it is a multiple of 3 or 9. | ## Multiplication RuleWhat is the total number of possible outcomes when a pair of coins is tossed? When you toss a coin you have two outcomes either heard or tails. Therefore in each case we have number of the event=2. Let E1 = toss first coin n(E1) = 2. E2 = toss second coin n(E2) = 2. These events are independent. So n(E) = n(E1) × n(E2) = 2 × 2 = 4 When a pair of coins is tossed, the total number of possible outcomes is 4. | ## PermutationsConsider arranging 3 different colored chairs: pink, blue, green. How many ways can this be done? The possible permutations are: pink,blue,green pink,green,blue blue,pink,green blue,green,pink green,pink,blue green,blue,pink There are six different arrangements. Hence 6 ways. |

## Addition Rule

In how many ways can a number be chosen from 1 to 31 such that it is a multiple of 3 or 9? Let E1 = multiples of 3 E1 = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30} n(E1) = 10 E2 = multiples of 9: E2 = {9, 18, 27} n(E2) = 3 Events E1 and E2 are mutually exclusive (there are no common outcomes). n(E) = n(E1) + n(E2) = 10+3 = 13 There are thirteen ways for a number to be chosen from 1 to 31 such that it is a multiple of 3 or 9.

## Multiplication Rule

What is the total number of possible outcomes when a pair of coins is tossed? When you toss a coin you have two outcomes either heard or tails. Therefore in each case we have number of the event=2. Let E1 = toss first coin n(E1) = 2. E2 = toss second coin n(E2) = 2. These events are independent. So n(E) = n(E1) × n(E2) = 2 × 2 = 4 When a pair of coins is tossed, the total number of possible outcomes is 4.

Link for the videos: http://www.youtube.com/watch?v=oQpKtm5TtxU

http://www.youtube.com/watch?v=vOsKK9NECtA

Lynn Itani