### 8.4 Expecting the Unexpected

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• 9. Rolling Dice

You roll three dice. What is the probability of rolling at least two of the same number? (See Example 3 and Example 4.)

• Worked-Out Solution

The probability that all three numbers are different is

So, the probability that at least two numbers are the same is

In other words, if you rolled 3 dice 1000 times, you would expect at least two numbers to be alike around 444 times.

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```             ___      _  __    _____   __    __
____      / _ \\   | |/ //  |  ___|| \ \\ / //
|    \\   / //\ \\  | ' //   | ||__    \ \/ //
| [] ||  |  ___  || | . \\   | ||__     \  //
|  __//  |_||  |_|| |_|\_\\  |_____||    \//
|_|`-`   `-`   `-`  `-` --`  `-----`      `
`-`
```
• 10. Rolling Dice

You roll five dice. What is the probability of rolling at least two of the same number? (See Example 3 and Example 4.)

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``` _    _     _____     ______   __   __   __   _
| \  / ||  |  ___||  /_   _//  \ \\/ // | || | ||
|  \/  ||  | ||__    `-| |,-    \ ` //  | '--' ||
| .  . ||  | ||__      | ||      | ||   | .--. ||
|_|\/|_||  |_____||    |_||      |_||   |_|| |_||
`-`  `-`   `-----`     `-`'      `-`'   `-`  `-`

```
• 11. Rolling Dice

You roll six dice. What is the probability of rolling at least one odd number? (See Example 3 and Example 4.)

• Worked-Out Solution

The probability that all six numbers are even is

So, the probability that at least one number is odd is

In other words, if you rolled 6 dice 1000 times, you would expect at least one number to be odd about 984 times.

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

```
• 12. Rolling Dice

You roll seven dice. What is the probability of rolling at least two of the same number? (See Example 3 and Example 4.)

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```  ______   _    _     _____     ______   ______
/_   _// | || | ||  |__  //   /_   _// |      \\
`-| |,-  | || | ||    / //     -| ||-  |  --  //
| ||   | \\_/ ||   / //__    _| ||_  |  --  \\
|_||    \____//   /_____||  /_____// |______//
`-`'     `---`    `-----`   `-----`  `------`

```
• 13. Drawing Marbles from a Bag

There are 26 marbles in a bag. The graph shows the color distribution of the marbles. Assume that you randomly draw one marble from the bag and you put the marble back before drawing another.

You randomly draw three marbles from the bag. What is the probability that none of the marbles are red? (See Example 3 and Example 4.)

• Worked-Out Solution

The probability that none of the marbles are red is

In other words, if you drew 3 marbles from the bag 1000 times, you would expect none of the marbles to be red about 455 times.

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

```
• 14. Drawing Marbles from a Bag

There are 26 marbles in a bag. The graph shows the color distribution of the marbles. Assume that you randomly draw one marble from the bag and you put the marble back before drawing another.

You randomly draw five marbles from the bag. What is the probability that none of the marbles are yellow? (See Example 3 and Example 4.)

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

```
• 15. Drawing Marbles from a Bag

There are 26 marbles in a bag. The graph shows the color distribution of the marbles. Assume that you randomly draw one marble from the bag and you put the marble back before drawing another.

You randomly draw four marbles from the bag. What is the probability that at least one of the marbles is green? (See Example 3 and Example 4.)

• Worked-Out Solution

The probability that none of the marbles are green is

So, the probability that at least one of the marbles is green is

In other words, if you drew 4 marbles from the bag 1000 times, you would expect at least one of the marbles to be green about 770 times.

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.

```           _    _     ______    _____     _____
____    | || | ||  /_   _//  |  ___||  /  ___||
|    \\  | || | ||  `-| |,-   | ||__   | // __
| [] ||  | \\_/ ||    | ||    | ||__   | \\_\ ||
|  __//   \____//     |_||    |_____||  \____//
|_|`-`     `---`      `-`'    `-----`    `---`
`-`
```
• 16. Drawing Marbles from a Bag

There are 26 marbles in a bag. The graph shows the color distribution of the marbles. Assume that you randomly draw one marble from the bag and you put the marble back before drawing another.

You randomly draw six marbles from the bag. What is the probability that at least one marble is blue? (See Example 3 and Example 4.)

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``` _    _      ___      _____    __   __  __    __