Conditional Probability

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# Conditional Probability
### Bora Jin

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## Material

🎥 Watch [Conditional Probability](https://warpwire.duke.edu/w/9RUGAA/)

- [Slides](https://sta198f2021.github.io/website/slides/week-02/w2-l04-condprob.html)

🎥 Watch [Simpson's Paradox](https://www.youtube.com/watch?v=sdas62v0iJU)

- [Slides](https://rstudio-education.github.io/datascience-box/course-materials/slides/u2-d16-simpsons-paradox/u2-d16-simpsons-paradox.html#1)

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## Today's Goal

- Define marginal, joint, and conditional probabilities, and calculate each “manually” and in a reproducible way
- Identify when events are independent
- Apply Bayes' theorem to examine COVID-19 test specificity

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## Quiz

**Q - Find the correct term, a mathematical notation with events** `$A$` **and** `$B$`, **and an example for each of the following descriptions.**

"The probability an event occurs regardless of values of the other event"

- Marginal probability

- `$P(A)$` or `$P(B)$`

- The probability that a randomly selected student in STA199 favors dogs over cats.

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## Quiz

**Q - Find the correct term, a mathematical notation with events** `$A$` **and** `$B$`, **and an example for each of the following descriptions.**

"The probability two or more events simultaneously occur"

- Joint probability

- `$P(A \cap B)$`

- The probability that a randomly selected student in STA199 favors dogs **and** their favorite movie genre is drama.

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## Quiz

**Q - Find the correct term, a mathematical notation with events** `$A$` **and** `$B$`, **and an example for each of the following descriptions.**

"The probability an event occurs given the other has occurred"

- Conditional probability

- `$P(A|B) = \frac{P(A \cap B)}{P(B)}$` or `$P(B|A)$`

- The probability that a randomly selected student in STA199 favors dogs **given** their favorite movie genre is drama.

- `$A \text{ given } B$` restricts our attention to `$B$` from the whole sample space `$\Omega$`.

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## Quiz

**Q - What is the multiplicative rule of probability?**

`$$P(A \cap B) = P(A|B)P(B)$$`

Comes directly from definition of conditional probability.

**Q - What is the law of total probability?**

`$$P(A) = P(A|B)P(B) + P(A|B^c)P(B^c) = P(A \cap B) + P(A \cap B^c)$$`

Let's verify it with a Venn Diagram!

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## Quiz

**Q - What are the independent events?**

- Knowing one event has occurred does not lead to any change in the probability we assign to another event.

- `$P(A|B) = P(A)$` or `$P(B|A) = P(B)$`

- `$P(A\cap B) = P(A)P(B)$` if `$A$` and `$B$` are independent.

- P(animal|movie) = P(animal)

- Knowing one's movie taste tells us nothing about his/her preference of dogs over cats.

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## Quiz

**Q - Are two disjoint events with positive marginal probability independent?**

`$P(A)P(B) > 0$` while `$P(A \cap B)=0$` and thus `$P(A \cap B) \neq P(A)P(B)$`.

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**Q - Are dying and abstaining from coffee independent events?**

Let `$A$` be the event that a man died and `$B$` be the event that a man was a non-coffee drinker.

.midi[

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

- `$P(A) = 9080/64821$`
- `$P(B) = 6477/64821$`
- `$P(A \cap B) = 1039/64821$`
--

- `$P(A)P(B) = 0.014 \approx P(A \cap B) = 0.016$`

--
They are independent!

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## Quiz

**Q - What is the probability that a man was a non-coffee drinker given he died?**

- `$P(A) = 9080/64821$`
- `$P(B) = 6477/64821$`
- `$P(A \cap B) = 1039/64821$`

`$P(B|A) = P(B \cap A) / P(A) = 1039 / 9080 = 0.114$`

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## Quiz

**Q - What is the probability that a man died given he was a non-coffee drinker?**

- `$P(A) = 9080/64821$`
- `$P(B) = 6477/64821$`
- `$P(A \cap B) = 1039/64821$`

`$P(A|B) = P(A \cap B) / P(B) = 1039 / 6477 = 0.160$`

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## Quiz

**Q - ** `$P(A|B)$` **and** `$P(B|A)$` **are not the same. Are they related in some way?**

Yes - According to Bayes' theorem,

`$$P(A|B) = \frac{P(B|A)P(A)}{P(B)}.$$`
If we know marginal probabilities and one conditional probability, we can always compute the other conditional probability!

This can be viewed as updating prior belief `$P(A)$` to posterior belief `$P(A|B)$` after seeing new information `$B$`.

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## Quiz

**Q - Let's recompute** `$P(A|B)$` **using Bayes' theorem.**

- `$P(A) = 9080/64821$`
- `$P(B) = 6477/64821$`
- `$P(B|A) = 1039/9080$`

`\begin{align}
P(A|B) &= \frac{P(B|A)P(A)}{P(B)} \\
&= \frac{1039/9080 \times 9080/64821}{6477/64821} \\
&= \frac{1039}{6477} = 0.160
\end{align}`

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## Quiz

**Q - What is Simpson's paradox?**

The effect where the inclusion of a third variable in the analysis can change the apparent relationship between the other two variables

**Q - Explain Simpson's paradox in UC Berkeley admission data in mathematical expressions.**

Overall, P(Admit|Female) `$<$` P(Admit|Male)

However, P(Admit|Dept, Female) `$\approx$` P(Admit|Dept, Male) or even P(Admit|Dept, Female) `$>$` P(Admit|Dept, Male)

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## Quiz

**Q - The effect of the function** `group_by()` **applies to all the subsequent operations. What function do we need to undo grouping?**

- `ungroup()`

- `summarize()` also peels off one layer of grouping by default; if there were multiple grouping variables, then the dataset would still be grouped even after summarize.

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# Questions?

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## Let's Practice Together!

Go to [AE 11: Conditional Probability](https://sta199-summer22.netlify.app/appex/ae11_BJ.html)

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## Bulletin

- Watch videos for [Prepare: May 27](https://sta199-summer22.netlify.app/prepare/week03_may27_BJ.html)

- Submit your `ae11`

- Exam 01 tomorrow!