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### 10.3 Professional Sports

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• 7. 3rd Down Play Balance

The graphs show 3rd down statistics for all of the regular season NFL games from 2000 through 2007. Plays within 2 minutes of the end of a half and all plays within field goal range are excluded. Do NFL coaches call more pass plays or run plays on 3rd down? Explain your reasoning. (See Example 3 and Example 4.)

• Worked-Out Solution

In general, NFL coaches call more pass plays than run plays on 3rd down. The only whole number value for number of yards to go for which a coach is more likely to call a run play than a pass play is 1 yard to go.

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```             ___       _____    ______
____      / _ \\    / ___//  /_   _//   ____
|    \\   | / \ ||   \___ \\   -| ||-   |    \\
| [] ||   | \_/ ||   /    //   _| ||_   | [] ||
|  __//    \___//   /____//   /_____//  |  __//
|_|`-`     `---`   `-----`    `-----`   |_|`-`
`-`                                     `-`
```
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• 8. 3rd Down Success

The graphs show 3rd down statistics for all of the regular season NFL games from 2000 through 2007. Plays within 2 minutes of the end of a half and all plays within field goal range are excluded. Describe the pattern in the "3rd Down Success" graph. (See Example 3 and Example 4.)

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

```
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• 9. Conversion Rates on 3rd Down

The graphs show 3rd down statistics for all of the regular season NFL games from 2000 through 2007. Plays within 2 minutes of the end of a half and all plays within field goal range are excluded. For what "to go" distances are the conversion rates for running and passing about equal on 3rd down? (See Example 3 and Example 4.)

• Worked-Out Solution

The conversion rates for running and passing are about equal on 3rd down for 5, 6, 7, 8, and 10 yards to go because the red and blue lines either overlap or are very close to one another for those values.

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```  ____       ___                _____    __   _
|  _ \\    / _ \\    ____     |  ___|| | || | ||
| |_| ||  | / \ ||  |    \\   | ||__   | '--' ||
| .  //   | \_/ ||  | [] ||   | ||__   | .--. ||
|_|\_\\    \___//   |  __//   |_____|| |_|| |_||
`-` --`    `---`    |_|`-`    `-----`  `-`  `-`
`-`
```
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• 10. Expecting a Pass Play or Run Play

The graphs show 3rd down statistics for all of the regular season NFL games from 2000 through 2007. Plays within 2 minutes of the end of a half and all plays within field goal range are excluded. With 2 yards to go on 3rd down, should the defense expect a pass play or a run play? Explain your reasoning. (See Example 3 and Example 4.)

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``` ______      ___     _    _      ___
|      \\   / _ \\  | |  | ||   / _ \\    ____
|  --  //  | / \ || | |/\| ||  / //\ \\  |    \\
|  --  \\  | \_/ || |  /\  || |  ___  || | [] ||
|______//   \___//  |_// \_|| |_||  |_|| |  __//
`------`    `---`   `-`   `-` `-`   `-`  |_|`-`
`-`
```
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• 11. Deciding Whether to Pass or Run

The graphs show 3rd down statistics for all of the regular season NFL games from 2000 through 2007. Plays within 2 minutes of the end of a half and all plays within field goal range are excluded. With 2 yards to go on 3rd down, should the coach call a pass play or a run play? Explain your reasoning. (See Example 3 and Example 4.)

• Worked-Out Solution

With 2 yards to go on 3rd down, the coach should call a run play because the conversion rate is higher for run plays on 3rd down with 2 yards to go. Also, the defense is probably expecting the coach to call a pass play.

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

```
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• 12. Passing Efficiency

The graphs show 3rd down statistics for all of the regular season NFL games from 2000 through 2007. Plays within 2 minutes of the end of a half and all plays within field goal range are excluded. How might a quarterback's passing efficiency influence a coach's play selection on 3rd down and 4 yards to go? (See Example 3 and Example 4.)

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

```
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• 13. Optimum Mix for Running and Passing

The graphs show 3rd down statistics for all of the regular season NFL games from 2000 through 2007. Plays within 2 minutes of the end of a half and all plays within field goal range are excluded. The optimum mix for running and passing occurs when the conversion rates are about equal. When this happens, the overall conversion rate will be greatest. Is the pass/run ratio at the optimum mix on 3rd down and 1 yard to go? Explain your reasoning. (See Example 3 and Example 4.)

• Worked-Out Solution

The pass/run ratio is not at the optimum mix on 3rd down and 1 yard to go. The optimum mix for running and passing occurs when the conversion rates are about equal. From the graph, the conversion for run plays is about 10% greater than the conversion rate for pass plays on 3rd down with 1 yard to go.

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

```
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• 14. Coaching Decision

The graphs show 3rd down statistics for all of the regular season NFL games from 2000 through 2007. Plays within 2 minutes of the end of a half and all plays within field goal range are excluded. Suppose you are the coach of an NFL team. Your team is facing a 3rd down with 5 yards to go. Would you call a pass play or a run play? Explain your reasoning. (See Example 3 and Example 4.)

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

```
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