Probability

Lecture 10

Dr. Elijah Meyer

Duke University
STA 199 - Spring 2023

February 15th, 2023

Checklist

– Exam-1 is over! Congratulations on your hard work

– Clone ae-09

– HW-2 coming FRIDAY

Feedback

– HW’s, labs, AE’s and exam; there are a lot of submissions to keep track of

– I wish there was a cheat sheet for all the appropriate formatting for our HWs and labs.

– Structure of ae’s (i.e. consistently not finishing ae material in class)

– I think that sometimes it would be helpful to do a quick review in the lecture of what we’ve learned the class before.

Where we are going

\(\checkmark\) Data Viz

– Probability

– Modeling Data

Goals

• have a working understanding of the terms probability, sample space, event, population and sample.

• compute probabilities of events from data

• create a contingency table using pivot_wider() and kable()

• use a contingency table to explore the relationship between two categorical variables.

Probability

– The probability of an event tells us how likely an event is to occur

• takes on values from 0 to 1

• the proportion of times the event would occur if it could be observed an infinite number of times

• our degree of belief an event will happen

An Event

– is the basic element to which probability is applied, e.g. the result of an observation or experiment

• Example: A is the event a student in STA 199 is a sophomore

• We use capital letters, e.g. A to denote events

The oppositve of the Event

– If A is the event a student in STA199 is a sophomore, what is the opposite?

• For any event A and its complement is \(A^c\)

• Can think about c as “not”

• Pr(A) + Pr(\(A^c\)) = 1

Sample Space

– A sample space is the set of all possible outcomes

• Each outcome in the sample space is disjoint or mutually exclusive meaning they can’t occur simultaneously

Sample Space

– The sample space for year in school is….?

• {Freshman, Sophmore, Junior, Senior}

• each item brackets is a distinct outcome

• The probability of the entire sample space is 1

Example

Suppose you are interested in the probability of landing on heads.

Define the following:

• Event (and compliment)

• Sample space

ae-09

More Complicated Probabilities

– And

– Or

– Conditional

Conditional Probabilities

– “Given an event has already happened…..”

– A | B

– Cuts up our contingency table

Work through

Work through

In Summary

– What is probability

– Event

– Sample Space

– Calculate probabilities in R

– And / Or / Conditional Probabilities