Introduction to Probability

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# Introduction to Probability
### Bora Jin

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<a href="https://introds.org" target="_blank">introds.org</a>
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## Material

🎥 Watch [Probability](https://warpwire.duke.edu/w/8RUGAA/)

- [Slides](https://sta198f2021.github.io/website/slides/week-02/w2-l03-prob.html)

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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.

**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"?**

- 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?**

- 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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- 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?

- 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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- 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?**

- "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"?**

- Set of all possible outcomes 
- {Freshman, Sophomore, Junior, Senior}
- `\(\Omega = A \cup A^c\)`
- 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: `\(A \cup B\)`

- 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?**

- 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?**

- [0, 1]

- The probability of the entire sample space is 1; `\(P(\Omega) = 1\)`

- 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|

.small[*Source:* https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5788283/]

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## Let's Practice Together!

Let `\(A\)` be the event that a man died and `\(B\)` be the event that a man was a non-coffee drinker. 
Calculate the following probabilities for a randomly selected person in the cohort:

.midi[

- `\(P(A)\)`

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= 9080/64821
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- `\(P(B)\)`

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= 6477/64821
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- `\(P(A \cap B)\)`

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= 1039/64821
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- `\(P(A \cup B)\)`

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= (9080 + 6477 - 1039)/64821
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- `\(P(A \cup B^c)\)`

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= (9080 + 58344 - 8041)/64821
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## Let's Practice Together!

Go to [AE 10: Introduction to Probability](https://sta199-summer22.netlify.app/appex/ae10_BJ.html)

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## Bulletin

- Watch videos for [Prepare: May 26](https://sta199-summer22.netlify.app/prepare/week03_may26_BJ.html)

- Submit your `ae10`

- HW02 released