If there is one prayer that you should pray/sing every day and every hour, it is the LORD's prayer (Our FATHER in Heaven prayer)
It is the most powerful prayer. A pure heart, a clean mind, and a clear conscience is necessary for it.
- Samuel Dominic Chukwuemeka

For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka

# Welcome to Probability

I greet you this day,
First: read the notes.
Second: view the videos.
Third: solve the questions/solved examples.
Fourth: check your solutions with my thoroughly-explained solutions.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me.
If you are my student, please do not contact me here. Contact me via the school's system. Thank you for visiting!!!

Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

## Objectives

Students will:

(1.) Discuss the meaning of probability.
(2.) Determine the probability of an event.
(3.) Compute the probabilities of events using the classical method.
(4.) Compute the probabilities of events using the empirical method.
(5.) Discuss scenarios related to probability including coins, dice, and cards among others.
(6.) Discuss the role of probability in life decisions including marriage among others.
(7.) Illustrate the sample spaces of events using probability tree diagrams.
(8.) Illustrate the sample spaces of events using Punnett squares.
(9.) Interpret the probabilities of events.
(10.) Determine the odds in favor of an event.
(11.) Determine the odds against an event.
(12.) Discuss the types of events.
(13.) Apply the rules of probability.
(14.) Solve applied problems involving probability.
(15.) Discuss conditional probability.
(16.) Solve applied problems involving conditional probability.
(17.) Discuss Bayes' theorem.

## Mathematicians who did extensive work on Probability

(1.) Gregor Mendel - Mendelian Laws of Genetics - Roman Catholic Monk - Austrian
(2.) Reverend Thomas Bayes - Bayes Theorem/Conditional Probability - British
(3.) Reginald Punnett - Punnett Square - British
(4.) Girolamo Cardano - Italian
(5.) Pafnuty Chebyshev - Chebyshev's Theorem - Russian
(6.) Andrey Markov - Markov Chain - Russian
(7.) Andrey Kolmogorov - Russian
(8.) Pierre de Fermat - France
(9.) Pierre Simon de Laplace - France
(10.) Blaise Pascal - France
***Honorable Mention - Evang. Samuel Dominic Chukwuemeka (SamDom For Peace) - The Humble Teacher

Okay, great people - please let me know whether I taught this topic well.
Did you learn anything from me? ☺☺☺

## Scenarios Related to Probability / Vocabulary Words

Bring it to English Language: probably, chance, odds, odds of winning, odds of losing, placing bets, gambling, sweepstakes, raffle tickets, door prizes, cards, coins, dice, result, success, failure, events, set, dependent, independent, possible, impossible, odds, odds in favor, odds against,

Bring it to Science: phenomenon, random, experiment, outcome, theoretical, classical, empirical, theorem, rule, law,

Bring it to Mathematics: probability, sample space, event space, mutually exclusive, mutually inclusive, inclusive, with replacement, without replacement, addition law, multiplication law, inequality

Check for Prior Knowledge
Ask students to list all the ways/terms/scenarios in which they have used the word, Probability or any word associated with Probability

(1.) Religion/Christianity: The probability that JESUS CHRIST will come again to judge all mankind is $1$ ($100\%$).

(2.) Meteorology: The National Weather Service predicted a $70\%$ chance of precipitation in Peachtree City, GA on the Saturday night of $3^{rd}$ February, $2018$

(3.) Gambling/Gaming: What are the odds in favor of winning the Powerball lottery?

(4.) Gambling/Gaming: What are the odds against winning the Mega Milions lottery?

(5.) Biology: The probability of seeing a pregnant man is $0$ $(0\%)$.

(6.) Statistics: It is generally known that one with a higher degree will earn more than another with a lower degree or no degree at all.
According to the Bureau of Labor Statistics, one of the reasons for varying wages is Credentials.
What is the probability that a randomly selected resident of the State of New Hampshire earns more than the minimum income for middle class according to CNBC given that the person has an advanced degree?
Did you notice the underlined, "given that"? What is the probability that an event will occur given that a prior event occurred?
This is known as Conditional Probability.

(7.) Criminology: Have you heard of a: true positive? true negative? false positive? false negative?
Is it possible for someone who is drug-free to test positive for drug-use? - Of course, it is possible.
Remember we are humans. Science helps us to avoid mistakes. With the advancement of science and technology, the "probability" of making those mistakes are small.
But, we still make mistakes because we are humans. We have limitations. Do you see another reason of using "probability"?
Is it also possible for someone who uses drugs to test negative for drug-use? - Of course, it is also possible.

and much more scenarios ...

## Introduction

Let us define and explain several terms that we shall use in this topic.

Deterministic Phenomenon: is a phenomenon that you can predict the outcome based on the information you know.
For example: Assume you went to the Walmart store at a certain location to buy oranges.

## References

Chukwuemeka, S.D (2016, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology. Retrieved from https://www.chukwuemekasamuel.com

Bennett, J. O., & Briggs, W. L. (2019). Using and Understanding Mathematics: A Quantitative Reasoning Approach. Pearson.

Black, Ken. (2012). Business Statistics for Contemporary Decision Making (7th ed.). New Jersey: Wiley

Blitzer, R. (2015). Thinking Mathematically (6th ed.). Boston: Pearson

Gould, R., & Ryan, C. (2016). Introductory Statistics: Exploring the world through data (2nd ed.). Boston: Pearson

Kozak, Kathryn. (2015). Statistics Using Technology (2nd ed.).

OpenStax, Introductory Statistics.OpenStax CNX. Sep 28, 2016. Retrieved from https://cnx.org/contents/30189442-6998-4686-ac05-ed152b91b9de@18.12

Sullivan, M., & Barnett, R. (2013). Statistics: Informed decisions using data with an introduction to mathematics of finance (2nd custom ed.). Boston: Pearson Learning Solutions.

Tan, S. (2015). Finite Mathematics for the Managerial, Life, and Social Sciences (11th ed.). Boston: Cengage Learning.

Triola, M. F. (2015). Elementary Statistics using the TI-83/84 Plus Calculator (5th ed.). Boston: Pearson

Triola, M. F. (2022). Elementary Statistics. (14th ed.) Hoboken: Pearson.

Weiss, Neil A. (2015). Elementary Statistics (9th ed.). Boston: Pearson

Authority (NZQA), (n.d.). Mathematics and Statistics subject resources. www.nzqa.govt.nz. Retrieved December 14, 2020, from https://www.nzqa.govt.nz/ncea/subjects/mathematics/levels/

CSEC Math Tutor. (n.d). Retrieved from https://www.csecmathtutor.com/past-papers.html

CDC. (2017, June 22). What is Sickle Cell Trait? Centers for Disease Control and Prevention. https://www.cdc.gov/ncbddd/sicklecell/traits.html

Centers for Disease Control and Prevention. (2020, December 14). What Is Sickle Cell Disease? Centers for Disease Control and Prevention. https://www.cdc.gov/ncbddd/sicklecell/facts.html

DLAP Website. (n.d.). Curriculum.gov.mt. https://curriculum.gov.mt/en/Examination-Papers/Pages/list_secondary_papers.aspx

Elementary, Intermediate Tests and High School Regents Examinations : OSA : NYSED. (2019). Nysedregents.org. https://www.nysedregents.org/

Free Jamb Past Questions And Answer For All Subject 2020. (2020, January 31). Vastlearners. https://www.vastlearners.com/free-jamb-past-questions/

GCSE Exam Past Papers: Revision World. Retrieved April 6, 2020, from https://revisionworld.com/gcse-revision/gcse-exam-past-papers

HSC exam papers | NSW Education Standards. (2019). Nsw.edu.au. https://educationstandards.nsw.edu.au/wps/portal/nesa/11-12/resources/hsc-exam-papers

NSC Examinations. (n.d.). www.education.gov.za. https://www.education.gov.za/Curriculum/NationalSeniorCertificate(NSC)Examinations.aspx

Papua New Guinea: Department of Education. (n.d.). www.education.gov.pg. Retrieved November 24, 2020, from http://www.education.gov.pg/TISER/exams.html

School Curriculum and Standards Authority (SCSA): K-12. Past ATAR Course Examinations. Retrieved December 10, 2021, from https://senior-secondary.scsa.wa.edu.au/further-resources/past-atar-course-exams

West African Examinations Council (WAEC). Retrieved May 30, 2020, from https://waeconline.org.ng/e-learning/Mathematics/mathsmain.html