Weekly Summary 3.1

University of the Potomac

Data Analytics

CBSC 520 Weekly Summary 3.1

In this week, I learned the topics about Probability Essentials like Addition rule, Conditional Probability and Multiplication rule, Equally Likely Events, Probability Distribution of a single variable, Summary Measures of a Probability Distribution, Conditional Mean and Variance and Introduction to Simulation. The addition rule states the probability of two events is the sum of the probability that either will happen minus the probability that both will happen. The multiplication rule states that the probability that AA and BB both occur is equal to the probability that BB occurs times the conditional probability that AA occurs given that BB occurs (Albright & Winston, 2017, Chapter 3).

In probability theory, to say that two events are independent means that the occurrence of one does not affect the probability that the other will occur. In other words, if events AA and BB are independent, then the chance of AA occurring does not affect the chance of BB occurring and vice versa.  A conditional mean is calculated much like a mean is, except you replace the probability mass function with a conditional probability mass function (Albright & Winston, 2017, Chapter 3).

Probability

A probability is a number between 0 and 1 that measures the likelihood that some event will occur. In probability, 0 shows event won’t occur and 1 shows certain occur. An event with probability greater than 0 and less than 1 involves uncertainty. The closer its probability is to 1, the more likely it is to occur (Albright & Winston, 2017, Chapter 3).

Example: If the chance of rain is 80%, then the probability of rain is 0.8. Similarly, if the odds against the Cavaliers winning are 3 to1, then the probability of the Cavaliers winning is 1/4 (or 0.25).

Conditional Probability

Probabilities are always assessed relative to the information currently available. When we have new information and other events are available, probabilities can change.

Formula: P(A∣B) = P (A and B)/P(B) here P(A|B) means “Event A given Event B”

Example: Drawing 2 Kings from a Deck

Event A is drawing a King first, and Event B is drawing a King second.

For the first card the chance of drawing a King is 4 out of 52 (there are 4 Kings in a deck of 52 cards): P(A) = 4/52. But after removing a King from the deck the probability of the 2nd card drawn is less likely to be a King (only 3 of the 51 cards left are Kings): P(B|A) = 3/51

And so: P (A and B) = P(A) x P(B|A) = (4/52) x (3/51) = 12/2652 = 1/221

So, the chance of getting 2 Kings is 1 in 221, or about 0.5%

Objective probability

Objective probability means an event will occur based on an analysis in which each measure is based on a recorded observation or a long history of collected data (Albright & Winston, 2017, Chapter 3).

For example, one could determine the objective probability that a coin will land “heads” up by flipping it 100 times and recording each observation. This would yield an observation that the coin landed on “heads” approximately 50% of the time. This is an example of objective probability.

Subjective probability

Subjective probability is a type of probability derived from an individual’s personal judgment or own experience about whether a specific outcome is likely to occur (Albright & Winston, 2017, Chapter 3)

An example of subjective probability is asking Manchester United FC fans, before the Soccer season starts, about the chances of Manchester United FC winning Premier league. While there is no absolute mathematical proof behind the answer to the example, fans might still reply in actual percentage terms, such as the MANU having a 55% chance of winning the title. Here is no calculation involved totally personal judgement exists.

Multiplication rule

The multiplication rule is a way to find the probability of two events happening at the same time. There are two multiplication rules. The general multiplication rule formula is: P(A ∩B) = P(A) P(B|A) and the specific multiplication rule is P(A and B) = P(A) * P(B). P(B|A) means “the probability of A happening given that B has occurred” (“Multiplication Rule Probability: Definition, Examples,” 2018).

Probability Distribution of a single variable

A probability distribution is a function that describes the likelihood of obtaining the possible values that a random variable can assume. In other words, the values of the variable vary based on the underlying probability distribution (“Understanding Probability Distributions,” 2018).

Conditional Mean

 A conditional mean is calculated much like a mean is, except you replace the probability mass function with a conditional probability mass function. And, a conditional variance is calculated much like a variance is, except you replace the probability mass function with a conditional probability mass function (“Conditional Means and Variances,” n.d.). 

Reference

Albright, S. C., & Winston, W. L. (2017). Business Analytics: Data Analysis & Decision Making. Boston, MA: Cengage Learning.

Conditional Probability. (n.d.). Retrieved from http://www.stat.yale.edu/Courses/1997-98/101/condprob.htm

Conditional Means and Variances. (n.d.). Retrieved from https://newonlinecourses.science.psu.edu/stat414/node/116/

Kenton, W. (2007, May 7). How the Addition Rule for Probabilities Works. Retrieved from https://www.investopedia.com/terms/a/additionruleforprobabilities.asp

Multiplication Rule Probability: Definition, Examples. (2018, June 2). Retrieved from https://www.statisticshowto.datasciencecentral.com/multiplication-rule-probability/

Understanding Probability Distributions. (2018, April 24). Retrieved from https://statisticsbyjim.com/basics/probability-distributions/