Counting Techniques: Permutations, Combinations, and Sample Space, Lecture notes of Probability and Statistics

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Counting Techniques

By: Engr. Marc Rivera Sample space, Events, Permutation, Combination

The set of all possible outcomes of a statistical experiment.

Sample Space

The sample space of possible outcomes when a coin is tossed. S = {head, tail} Note: in some expt. It is helpful to list the elements by means of tree diagram.

Example

An expt. consists of flipping a coin and then flipping it a second time if a head occurs. If a tail occurs on the first flip, then a die is tossed once. What are the possible sample space?

Example

S = {HH,HT,T1,T2,T3,T4,T5,T6}

Answer

Suppose that three items are selected at random from a manufacturing process. Each item is inspected and classified defective, D, or nondefective, N. List the all possible sample points by tree diagram.

Example

Used to describe sample spaces with a large or infinite number of sample points.

Statement or Rule Method

The possible outcomes of an experiment are the set of cities in the world with a population over 1 million. S = {x | x is a city with a population over 1 million} It reads “S is the set of all x such that x is a city with a population over 1 million.”

Example

Answer

S = {(x, y) | x 2

• y 2 ≤ 4}

Events

II. Events An event is a subset of a sample space.

Answer

S = {TT,TH,HT,HH}

E = {TH,HT}

Example

Given the sample space S = {t | t ≥ 0}, where t is the life in years of a certain electronic component, then the event A that the component fails before the end of the fifth year is:

Example

Let A be the event of detecting a microscopic organism by the naked eye in a biological experiment, then

Answer

A = ∅ or null set.