# Counting & Probability

## 1. 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. ## 2. Combinations

A combination is the choice of r things from a set of n things without replacement and where order does not matter.

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)!]

Example: Five people are in a club and three are going to be in the planning committee to determine how many different ways this committee can be created we use our combination formula as follows: ## 3. Permutations

A permutation is the choice of r things from a set of n things without replacement and where the order matters.

Example: In how many ways can a supermarket manager display 5 brands of cereals in 3 spaces on a shelf? 5P3 = (5!) / (5-3)! = 60

The manager can arrange them in 60 ways ## 4. Counting

An efficient way of counting is necessary to handle large masses of statistical data (e.g. the level of inventory at the end of a given month, or the number of production runs on a given machine in a 24 hour period, etc.), and for an understanding of probability.

Number of Outcomes of an Event: As an example, we may have an event E defined as E = "day of the week" We write the "number of outcomes of event E" as n(E).

Addition Rule: Let E1 and E2 be mutually exclusive events (i.e. there are no common outcomes). Let event E describe the situation where either event E1 or event E2 will occur.The number of times event E will occur can be given by the expression:n(E) = n(E1) + n(E2)

Where:

n(E) = Number of outcomes of event E

n(E1) = Number of outcomes of event E1

n(E2) = Number of outcomes of event E2

Multiplication Rule:Now consider the case when two events E1 and E2 are to be performed and the events E1 and E2 are independent events i.e. one does not affect the others outcome.

Example:Say the only clean clothes you've got are 2 t-shirts and 4 pairs of jeans. How many different combinations can you choose? 4 pairs of jeans. How many different combinations can you choose?

We can think of it as follows: We have 2 t-shirts and with each t-shirt we could pick 4 pairs of jeans. Altogether there are

2×4=8 possible combinations.

We could write

E1 = "choose t-shirt" and

E2 = "choose jeans"

## 5. Factorial Notation

For the following sections on counting, we need a simple way of writing the product of all the positive whole numbers up to a given number. We use factorial notation for this.

Definition of n!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)

Example: 5! = 5 × 4 × 3 × 2 × 1 = 120

What Is Probability?