# Counting & Probability

## 1. Basic Probability

**Probability**is 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.

Definitions:

1. Experiment: is a situation involving chance or probability that leads to results.

2. Outcome: is the result of a single trial of an experiment.

3. Event: is one or more outcomes of an experiment.

4. Probability of an event happening is: (number of favorable cases)/(number of possible cases)

__Experiment 1:__(Spinner)

A spinner has 4 equal sectors colored yellow, blue, green and red. After spinning the spinner, what is the probability of landing on each color?

**Outcomes**: The possible outcomes of this experiment are yellow, blue, green, and red.

**Probabilities:** P(yellow) = 1/4, P(blue) = 1/4, P(green) = 1/4, P(red) = 1/4.

__Experiment 2:__ (Dice)

A single 6-sided die is rolled. What is the probability of each outcome? What is the probability of rolling an even number? of rolling an odd number?

**Outcomes:** The possible outcomes of this experiment are 1, 2, 3, 4, 5 and 6.

**Probabilities:** P(1) = 1/6, P(2) = 1/6, P(3) = 1/6, P(4) = 1/6, P(5) = 1/6, P(6) = 1/6, P(even) = 3/6 = 1/2, P(odd) = 3/6 = 1/2

__Experiment 3:__ (Jar of Marbles)

A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a red marble? a green marble? a blue marble? a yellow marble?

**Outcomes:** The possible outcomes of this experiment are red, green, blue and yellow.

**Probabilities:** P(red) = 6/22 = 3/11, P(green) = 5/22, P(blue) = 8/22 = 4/11, P(yellow) = 3/22.

*Conclusion:*

In Experiments 1 & 2 the probability of each outcome is always the same. However, in experiment 3, the outcomes are not equally to occur. You are more likely to choose a blue marble than any other color. You are least likely to choose a yellow marble.

## 2. Counting

__Definitions:__

*E = *An Event

*n(E) *= Number of outcomes of event *E*

__Addition Rule:__

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

Let event *E* describe the situation where either event *E*1 *or* event *E*2 will occur.

The number of times event *E* will occur can be given by the expression:

*n*(

*E*) =

*n*(

*E*1) +

*n*(

*E*2)

where

*n*(*E*) = Number of outcomes of event *E*

* n*(*E*1) = Number of outcomes of event *E*1

* n*(*E*2) = Number of outcomes of event *E*2

__Multiplication Rule:__

Suppose that event *E*1 can result in any one of *n*(*E*1) possible outcomes; and for each outcome of the event*E*1, there are *n*(*E*2) possible outcomes of event *E*2.

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

That is, if event *E* is the event that both *E*1 and *E*2 **must** occur, then

*n*(*E*) = *n*(*E*1) × *n*(*E*2)

__Example:__

What is the total number of possible outcomes when a pair of coins is tossed?

__Answer:__

The events are described as:

*E*1 = toss first coin (2 outcomes, so *n*(*E*1) = 2.)

*E*2 = toss second coin (2 outcomes, so *n*(*E*2) = 2.)

They are independent, since neither toss affects the outcome of the other toss.

So *n*(*E*) = *n*(*E*1) × *n*(*E*2) = 2 × 2 = 4

[We could list the outcomes: HH HT TH TT].

## 3. Factorial Notation

** n factorial** is defined as the product of all the integers from 1 to

*n*(the order of multiplying does not matter) .

We write "*n* factorial" with an exclamation mark as follows: n!

*n*! = (

*n*)(

*n*− 1)(

*n*− 2)...(3)(2)(1)

The value of 0! is 1. (It is a convention)

Let us take an example:

4! = 4 × 3 × 2 × 1 = 24

## 4. Combinations

*n*objects taken

*r*at a time is a selection which does not take into account the arrangement of the objects. That is, the order is not important.

The number of ways (or **combinations**) in which *r* objects can be selected from a set of *n* objects, where repetition is **not** allowed, is denoted by nCr = (n!) / [r! (n-r)!], or:

## 5. Permutations

Consider 4 students walking toward their school entrance. How many different ways could they arrange themselves in this side-by-side pattern?

The number of different arrangements is 4! = (4)(3)(2)(1) = 24. There are 24 different arrangement, or permutations, of the four students walking side-by-side.

The notation for a permutation is: *n***P***r. *(The symbol nPr is the number of permutations formed from n objects taken r at a time)

The formula for a permutation is:

5P3 = (5!) / (5-3)! = 60

The manager can arrange them in 60 ways

However, in the above example, repetitions are not allowed!

**What if we had repetitions?**

In general, repetitions are taken care of by dividing the permutation by the factorial of the number of objects that are identical.

The equation for permutations with repetitions is:

In how many ways can the six letters of the word "mammal" be arranged in a row?

The word mammal consists of 3 "m"s and 2 "a"s, therefore we use the above formula,

(6!) / [(3!)(2!)] = 60

The six letters of the word mammal can be arranged in 60 ways.