Probability Theory: Sample Spaces, Events, and Rules, Lecture notes of Data Analysis & Statistical Methods

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Lecture 2

PROBABILITY

MathEng3-M (Engineering Data Analysis)

Outline of Discussion (Lecture 2)

➢ Sample Spaces and Events

➢ Counting Sample Points

➢ Set Theory

➢ Rules of Probability

Random Experiment

An experiment that can result in different

outcomes, even though it is repeated in the

same manner every time, is called a random

experiment.

Random Experiment

Sample Spaces and Events (Sample Space)

Sample Space

❑ In the study of statistics, we are concerned with the

presentation and interpretation of chance outcomes that occur

in a planned study or scientific investigation.

❑ For example, we can record the number of accidents that

occur monthly at the intersection of two streets, hoping to

justify an installation of a traffic signal.

❑ The statistician dealing with either experimental data,

representing counts or measurements or perhaps with

categorical data can be classified according to some criterion.

❑ Observation is any recording of information, whether it be

numerical or categorical.

Sample Space

Example 1:

Consider the experiment of tossing a die. If we are interested in

the number that shows on the top face, the sample space would

be,

S 1 = {1, 2, 3, 4, 5, 6}

If we are interested only in whether the number showing off is

even or odd, the sample space would be,

S 2 = {𝑒𝑣𝑒𝑛, 𝑜𝑑𝑑}

Sample Space

Example 2:

An experiment 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 is the sample space for the scenario?

Flip H H T T 1 2 3 4 5 6 Sample space: 𝑆 = {𝐻𝐻, 𝐻𝑇, 𝑇1, 𝑇2, 𝑇3, 𝑇4, 𝑇5, 𝑇6} 1 st outcome 2 nd outcome

Sample Space

Sample spaces with a large or infinite number of sample points

are best described by a statement or a rule.

Sample spaces with a large or infinite number of sample points

are best described by a statement or a rule.

Example: if the possible outcomes of an experiment are the set

of cities in the Philippines with a population over 100 , 000 , the

sample space can be written as:

𝑆={𝑥|𝑥 is a city with a population over 100,000}

Sample Spaces and Events (Events)

Events

Example 4:

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 component fails

before the fifth year is the subset,

Events ➢ From this, you can conclude that an event may be a subset that includes the entire sample space S, or a subset of S called the null set, denoted by Ø, which contains no elements at all. ➢ For example, if A is the event of accurate measurements by the naked eye in a surveying party, then 𝐴=∅. ➢ Also, if a sample space 𝐵={𝑥|𝑥 is an even factor of 7 }, then B must be a null set. (Why?)

Events DEFINITION: The intersection of two events A and B, denoted by 𝐴∩𝐵 is the event containing all elements that are common to A and B.

Try to solve the following:

a. Let P be the event that a person selected at random while dining at a cafeteria is a taxpayer, and let Q be the event that the person is over 65 years of age. Find the event 𝑃∩𝑄. b. Let 𝑀={𝑎, 𝑒, 𝑖, 𝑜, 𝑢} and 𝑁={𝑟, 𝑠, 𝑡}. Find 𝑀∩𝑁.