### 8.2 Estimating Likelihood

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• 1. Finding a Theoretical Probability

A dodecahedron die has 12 sides numbered 1 through 12. You roll a dodecahedron. Find the probability of rolling a 6. (See Example 1.)

• Worked-Out Solution

There are 12 possible outcomes. Of these, only 1 is favorable.

"Dice have been used throughout Asia since before recorded history; the oldest known dice were excavated as part of a 5000-year-old backgammon set at the Burnt City, an archeological site in south-eastern Iran. Other excavations from ancient tombs in the Indus Valley civilization indicate a South Asian origin." (Source: Wikipedia)

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```  ____       ___     ______      ___     _    _
|  _ \\    / _ \\  |      \\   / _ \\  | \  / ||
| |_| ||  | / \ || |  --  //  / //\ \\ |  \/  ||
| .  //   | \_/ || |  --  \\ |  ___  ||| .  . ||
|_|\_\\    \___//  |______// |_||  |_|||_|\/|_||
`-` --`    `---`   `------`  `-`   `-` `-`  `-`

```
• 2. Finding a Theoretical Probability

A dodecahedron die has 12 sides numbered 1 through 12. You roll a dodecahedron. Find the probability of rolling an 11. (See Example 1.)

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

```
• 3. Finding a Theoretical Probability

A dodecahedron die has 12 sides numbered 1 through 12. You roll a dodecahedron. Find the probability of rolling a number less than 9. (See Example 1.)

• Worked-Out Solution

There are 12 possible outcomes. Of these, only 8 are less than 9.

A dodecahedron is one of the five Platonic solids. The other four Platonic solids are tetrahedron (4 sides), cube (6 sides), octahedron (8 sides), and icosahedron (20 sides).

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

```
• 4. Finding a Theoretical Probability

A dodecahedron die has 12 sides numbered 1 through 12. You roll a dodecahedron. Find the probability of rolling a multiple of 4. (See Example 1.)

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``` ______      ___     _    _      ___     _    _
|      \\   / _ \\  | |  | ||   / _ \\  | \  / ||
|  --  //  / //\ \\ | |/\| ||  | / \ || |  \/  ||
|  --  \\ |  ___  |||  /\  ||  | \_/ || | .  . ||
|______// |_||  |_|||_// \_||   \___//  |_|\/|_||
`------`  `-`   `-` `-`   `-`   `---`   `-`  `-`

```
• 5. Finding a Theoretical Probability

A dodecahedron die has 12 sides numbered 1 through 12. You roll a dodecahedron. Find the probability of rolling an odd number. (See Example 1.)

• Worked-Out Solution

There are 12 possible outcomes. Of these, half are odd and half are even.

A dodecahedron is one of the five Platonic solids. The other four Platonic solids are tetrahedron (4 sides), cube (6 sides), octahedron (8 sides), and icosahedron (20 sides).

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

```
• 6. Finding a Theoretical Probability

A dodecahedron die has 12 sides numbered 1 through 12. You roll a dodecahedron. Find the probability of rolling a prime number. (See Example 1.)

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

```
• 7. Raffle

A charity sells 1000 tickets for a raffle. There is a grand prize of \$200 and 4 other prizes of \$50. You buy one ticket. (See Example 1.)

1. What is the probability that you win the grand prize?
2. What is the probability that you win a prize?
• Worked-Out Solution
1. There are 1000 possible outcomes. Of these, only 1 is your ticket.

2. There are 1000 possible outcomes. Of these, only 5 are winning tickets.

A raffle is a type of game often held to raise money for a specific event or charity. Tickets are randomly drawn from a group to determine the winner of each prize.

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```           __   __   _    _      ___      ____
___    \ \\/ // | || | ||   / _ \\   |  _ \\
/   ||   \ ` //  | || | ||  | / \ ||  | |_| ||
| [] ||    | ||   | \\_/ ||  | \_/ ||  | .  //
\__ ||    |_||    \____//    \___//   |_|\_\\
-|_||    `-`'     `---`     `---`    `-` --`
`-`
```
• 8. Lottery

The table shows the payouts for the 600,000 people who played the lottery yesterday. You randomly choose one person who played the lottery yesterday. Find the probability that the person is in each payout group. (See Example 2.)

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```  ______   _    _    __   _    _    _      _____