8.4 Expecting the Unexpected

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• 23. Bayes' Theorem

For any two events with probabilities greater than 0,

You have the following information about students at a college.

• 49% of the students are male.

• 11% of the students are nursing majors.

• 9% of the nursing majors are male.

What is the probability that a student is a nursing major given that the student is male?

• Worked-Out Solution

Event 1 is "the student is a nursing major."

Event 2 is "the student is male."

Event 2, given event 1 is "the nursing major is male."

A set diagram might help you see what is going on. In the diagram, you can see that 9% of the 11% of the student population are male nursing majors. If you are given that the selected student is male, then you have restricted your total population to the 49% of the students who are male. So, given that a student is male, the probability that he is a nursing major is

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```  ____     _    _    ______      ___      ______
|  _ \\  | || | || |      \\   / _ \\   /_____//
| |_| || | || | || |  --  //  | / \ ||  `____ `
| .  //  | \\_/ || |  --  \\  | \_/ ||  /___//
|_|\_\\   \____//  |______//   \___//   `__ `
`-` --`    `---`   `------`    `---`    /_//
`-`
```
• 24. Bayes' Theorem

For any two events with probabilities greater than 0,

You have the following information about students at a college.

• 51% of the students are female
• 10% of the students are history majors
• 60% of the history majors are female

What is the probability that a student is a history major given that the student is female?

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```  ____       ___    __    __    _____     _____
|  _ \\    / _ \\  \ \\ / //  |  ___||  / ____||
| |_| ||  | / \ ||  \ \/ //   | ||__   / //---`'
| .  //   | \_/ ||   \  //    | ||__   \ \\___
|_|\_\\    \___//     \//     |_____||  \_____||
`-` --`    `---`       `      `-----`    `----`

```
• 25. Bayes' Theorem

For any two events with probabilities greater than 0,

You have the following information about voters in a local mayoral election.

• 61% of voters were registered Republican.
• 53% of voters voted Republican.
• 86% of voters who voted Republican were registered Republican.

What is the probability that a voter voted Republican given that the voter was registered Republican?

• Worked-Out Solution

Event 1 is "the voter voted Republican."

Event 2 is "the voter is a registered Republican."

Event 2, given event 1 is "the person who voted Republican is a registered Republican."

A set diagram might help you see what is going on. In the diagram, you can see that 86% of the 53% of the people who voted Republican are registered Republicans. If you are given that the selected voter is a registered Republican, then you have restricted your total population to the 61%. So, given that a person is a registered Republican, the probability that the person voted Republican is

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``` _____      _____    _    _     ______    _____
|  __ \\   |  ___|| | |  | ||  /_   _//  |__  //
| |  \ ||  | ||__   | |/\| ||   -| ||-     / //
| |__/ ||  | ||__   |  /\  ||   _| ||_    / //__
|_____//   |_____|| |_// \_||  /_____//  /_____||
-----`    `-----`  `-`   `-`  `-----`   `-----`

```
• 26. Bayes' Theorem

For any two events with probabilities greater than 0,

You have the following information about voters in a local congressional election.

• 74% of voters were registered Democrat
• 62% of voters voted Democrat
• 79% of voters who voted Democrat were registered Democrat

What is the probability that a voter voted Democrat given that the voter was registered Democrat?

These comments are not screened before publication. Constructive debate about the information on this page is welcome, but personal attacks are not. Please do not post comments that are commercial in nature or that violate copyright. Comments that we regard as obscene, defamatory, or intended to incite violence will be removed. If you find a comment offensive, you may flag it.
```__    __   _    _     ____      _____     _  __