Introduction to ProbabilityBora Jin1 / 19

Material

🎥 Watch Probability

Slides

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Today's Goal

Understand population, sample, event, sample space, and probability.
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.

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Quiz

One goal of statistics is to answer a research question, by making inferences about a population based on data in one or more samples.

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Quiz

One goal of statistics is to answer a research question, by making inferences about a population based on data in one or more samples.

Q - What is a research question?

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Quiz

One goal of statistics is to answer a research question, by making inferences about a population based on data in one or more samples.

Q - What is a research question?

What we want to learn and what we are curious about. For example:

How likely is it for patients with early Parkinson's disease to experience a serious movement disorder within 5 years after the first diagnosis?
Does the average amount of caffeine vary by vendor in 12 oz. cups of coffee at Duke coffee shops?
How popular is the president among college students?

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Quiz

Q - What is a "population"?

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Quiz

Q - What is a "population"?

Entire group we are interested in studying
Parameter: a numerical quantity derived from the population (almost always unknown)
If we had data from every unit in the population, we could just calculate population parameters and be done!

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Quiz

Q - What is a "population"?

Entire group we are interested in studying
Parameter: a numerical quantity derived from the population (almost always unknown)
If we had data from every unit in the population, we could just calculate population parameters and be done!

Q - What is a "sample"?

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Quiz

Q - What is a "population"?

Entire group we are interested in studying
Parameter: a numerical quantity derived from the population (almost always unknown)
If we had data from every unit in the population, we could just calculate population parameters and be done!

Q - What is a "sample"?

The group we have collected data from (a subset of our population of interest)
Statistic: a numerical quantity derived from a sample
Usually we have to settle with a sample and draw conclusions from it because the population is just too big!

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Quiz

Q - What is a good sample and why is it important?

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Q - What is a good sample and why is it important?

A sample that represents the target population well (has similar characteristics as the population) is a good sample
Representativeness is what makes the conclusions and inferences from the sample generalizable and valid to the whole population
Similar to tasting a spoonful of soup while cooking to make an inference about the entire pot

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Quiz

Q - Identify the population and potential samples to answer the following questions.

How likely is it for patients with early Parkinson's disease to experience a serious movement disorder within 5 years after the first diagnosis?

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Quiz

Q - Identify the population and potential samples to answer the following questions.

How likely is it for patients with early Parkinson's disease to experience a serious movement disorder within 5 years after the first diagnosis?

population: all patients with early Parkinson's disease
sample: patients with early Parkinson's disease who agree to be followed up from Duke Hospital

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Quiz

Q - Identify the population and potential samples to answer the following questions.

Does the average amount of caffeine vary by vendor in 12 oz. cups of coffee at Duke coffee shops?

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Quiz

Q - Identify the population and potential samples to answer the following questions.

Does the average amount of caffeine vary by vendor in 12 oz. cups of coffee at Duke coffee shops?

population: all 12 oz. cups of coffee at Duke coffee shops
sample: randomly selected 50 cups of coffee (12 oz.) by each vendor at Duke coffee shops

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Quiz

Q - Identify the population and potential samples to answer the following questions.

How popular is the president among college students?

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Quiz

Q - Identify the population and potential samples to answer the following questions.

How popular is the president among college students?

population: all college students
sample: randomly selected college students who share their opinions about the president

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Quiz

In order to draw principled conclusions from data, we rely on a formal probabilistic framework that allows us to quantify uncertainty.

Q - In probability theory, what is a term indicating the result of an observation or experiment?

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Quiz

In order to draw principled conclusions from data, we rely on a formal probabilistic framework that allows us to quantify uncertainty.

Q - In probability theory, what is a term indicating the result of an observation or experiment?

"Event"
Usually denoted by capital letters
$A$ is the event that a student in STA199 is a sophomore.
$A^{c}$ ( $A$ complement ) is the event that $A$ is not true; a student in STA199 is not a sophomore.

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Quiz

Q - What is a "sample space"?

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Q - What is a "sample space"?

Set of all possible outcomes
{Freshman, Sophomore, Junior, Senior}
$Ω = A \cup A^{c}$
Varies by research questions

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Quiz

Q - Say in words the following math expressions.

$A$ is the event that a student in STA199 is a sophomore. $B$ is the event that a student in STA199 is a senior.

Union: $A \cup B$

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Quiz

Q - Say in words the following math expressions.

$A$ is the event that a student in STA199 is a sophomore. $B$ is the event that a student in STA199 is a senior.

Union: $A \cup B$
Intersection: $A \cap B$

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Quiz

Q - Say in words the following math expressions.

$A$ is the event that a student in STA199 is a sophomore. $B$ is the event that a student in STA199 is a senior.

Union: $A \cup B$
Intersection: $A \cap B$
- Is it possible? If not, we say $A$ and $B$ are mutually exclusive or disjoint.

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Quiz

Q - Say in words the following math expressions.

$A$ is the event that a student in STA199 is a sophomore. $B$ is the event that a student in STA199 is a senior.

Union: $A \cup B$
Intersection: $A \cap B$
- Is it possible? If not, we say $A$ and $B$ are mutually exclusive or disjoint.
Complement: $B^{c}$

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Quiz

Q - What is the "probability" of an event?

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Quiz

Q - What is the "probability" of an event?

How likely an event is to occur
Interpretations (1) vs. (2)
- (1) The proportion of times the event would occur if it could be observed an infinite number of times.
- (2) Degree of belief an event will happen

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Quiz

Q - Probability Rules?

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Q - Probability Rules?

[0, 1]

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Q - Probability Rules?

[0, 1]
The probability of the entire sample space is 1; $P (Ω) = 1$

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Q - Probability Rules?

[0, 1]
The probability of the entire sample space is 1; $P (Ω) = 1$
Complement rule: $P (A) + P (A^{c}) = 1$

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Quiz

Q - Probability Rules?

[0, 1]
The probability of the entire sample space is 1; $P (Ω) = 1$
Complement rule: $P (A) + P (A^{c}) = 1$
Additive rule: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$
- Avoiding double-counting
- If $A$ and $B$ are disjoint, $P (A \cap B) = 0$ and thus $P (A \cup B) = P (A) + P (B)$

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Questions?

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Let's Practice Together!

For 64,821 men enrolled in the European Prospective Investigation into Cancer and Nutrition after a mean follow-up of 16.4 years

	Did not die	Died
Does not drink coffee	5438	1039
Drinks coffee occasionally	25369	4440
Drinks coffee regularly	24934	3601

Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5788283/

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