Probability Review

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# Probability Review
## Dr. Mine Dogucu
Examples from [bayesrulesbook.com](https://bayesrulesbook.com)

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# Probability Review

<p style="font-size: small">
Data from <a href ="https://gssdataexplorer.norc.org"> General Social Survey</a>
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`\(P(\text{belief in afterlife})\)` = ?
`\(P(\text{belief in afterlife and taken a college science class})\)` = ?  
`\(P(\text{belief in afterlife given taken a college science class})\)` = ?

Calculate these probabilities and write them using correct notation. Use `\(A\)` for belief in afterlife and `\(B\)` for college science class.

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### Marginal Probability

<p style="font-size: small">
Data from <a href ="https://gssdataexplorer.norc.org"> General Social Survey</a>
</p>

`\(P(\text{belief in afterlife})\)` = ?  
`\(P(A) = \frac{6424}{7895}\)`

`\(P(A)\)` represents a __marginal probability__. So do `\(P(B)\)`, `\(P(A^C)\)` and `\(P(B^C)\)`. In order to calculate these probabilities we could only use the values in the margins of the contingency table, hence the name.

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### Joint Probability

<p style="font-size: small">
Data from <a href ="https://gssdataexplorer.norc.org"> General Social Survey</a>
</p>

`\(P(\text{belief in afterlife and taken a college science class})\)` = ? 
`\(P(A \text{ and } B) = P(A \cap B) = \frac{2702}{7895}\)`

`\(P(A \cap B)\)` represents a __joint probability__. So do `\(P(A^c \cap B)\)`, `\(P(A\cap B^c)\)` and `\(P(B^c\cap B^c)\)`.

Note that `\(P(A\cap B) = P(B\cap A)\)`. Order does _not_ matter.

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### Conditional Probability

<p style="font-size: small">
Data from <a href ="https://gssdataexplorer.norc.org"> General Social Survey</a>
</p>

`\(P(\text{belief in afterlife given taken a college science class})\)` = ?
`\(P(A \text{ given } B) = P(A | B) = \frac{2702}{3336}\)`

`\(P(A|B)\)` represents a __conditional probability__. So do `\(P(A^c|B)\)`, `\(P(A | B^c)\)` and `\(P(A^c|B^c)\)`. In order to calculate these probabilities we would focus on the row or the column of the given information. In a way we are _reducing_ our sample space to this given information only.

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## Note on conditional probability

`\(P(\text{attending every class | getting an A}) \neq\)` `\(P(\text{getting an A | attending every class})\)`

The order matters!

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## Complement of an Event

`\(P(A^C)\)` is called __complement__ of event A and represents the probability of selecting someone that does not believe in afterlife.