Defining Probability

- Lesson 8.8 -

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Khris Davis

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Khris Davis

Chris Davis

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Khris Davis

Chris Davis

Today’s Key Analysis (#1)

How improbable were the MLB records set by the two Khris/Chris Davis’s?

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Lesson 8.8

Guided Notes

Handout: skewthescript.org/5-1

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Topics

• Law of large numbers
• Simple definition of probability
• Properties of probabilities
• Probability models

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Topics

• Law of large numbers
• Simple definition of probability
• Properties of probabilities
• Probability models

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Law of Large Numbers

Law of large numbers: After many trials, the relative frequency of outcomes will approach their probability.

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Applet (coin flips)

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http://digfir-published.macmillanusa.com/stats_applet/stats_applet_10_prob.html

Source: Digital First project from Bedford, Freeman, & Worth publishers

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Source: Digital First project from Bedford, Freeman, & Worth publishers

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Being “due for one”

You have a fair coin. You flip it 10 times. In which situation is getting tails (T) on your next flip more likely?

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Situation 1: HTTHHTHTHT

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Situation 2: HTTHHHHHHH

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Being “due for one”

You have a fair coin. You flip it 10 times. In which situation is getting tails (T) on your next flip more likely?

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Situation 1: HTTHHTHTHT

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Situation 2: HTTHHHHHHH

It’s equally likely! The probability doesn’t change because of previous results.

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Being “due for one”

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Khris Davis

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Khris Davis

Chris Davis

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Khris Davis

2015

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Batting Average: 0.247

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Home Runs: 27

Source: Baseball Reference

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Khris Davis

2015

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Batting Average: 0.247

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Home Runs: 27

24.7% of his at-bats ended in hits (rather than outs)

Source: Baseball Reference

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

2015

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Batting Average: 0.262

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Home Runs: 47

Source: Baseball Reference

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

2015

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Batting Average: 0.262

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Home Runs: 47

Source: Baseball Reference

Led the MLB

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

2015

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Batting Average: 0.262

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Home Runs: 47

Source: Baseball Reference

Led the MLB

Led the MLB in neck size (probably)

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

2018

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Batting Average:

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Home Runs:

Source: Baseball Reference

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

2018

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Batting Average: 0.168

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Home Runs: 16

Source: Baseball Reference

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

2018

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Batting Average: 0.168

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Home Runs: 16

Source: Baseball Reference

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0,

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2,

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1,

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1,

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0,

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1,

Sept. 14, 2018

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0,

Sept. 14, 2018

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0,

Sept. 14, 2018

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

“He’s due for a hit”

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

0,

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

0, 0,

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

0, 0, 0,

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

0, 0, 0, 0,

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

0, 0, 0, 0,

“He’s due for a hit”

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

0, 0, 0, 0, 0,

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

0, 0, 0, 0, 0, 0,

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

17 straight hitless games

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

Number of Hits per Game

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3

April 13, 2019

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

54 straight at-bats without a hit

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3

April 13, 2019

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

54 straight at-bats without a hit

MLB Record

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3

April 13, 2019

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Chris Davis

54 straight at-bats without a hit

MLB Record

May 2018

0, 2, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1…

Sept. 2018

1, 0, 0, 0, 0, 0

Sept. 14, 2018

April 2019

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3

April 13, 2019

MLB modern-era began in 1900! All-time records are rare.

A Tale of Two Khris/Chris Davis’s

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Chris asks for the ball

Teammates celebrate

After the hit that broke the streak…

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Unlike the coin flips, Chris Davis’s chance at getting a hit isn’t the same for every at-bat. However, the “due for one” logic is still flawed:

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• Repeated failure doesn’t indicate a higher chance of success on a future attempt. If anything, it may indicate your original estimate of the probability of success was already too high.

“Due for one” myth

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Being “due for one”

A Tale of Two Khris/Chris Davis’s

Khris Davis

Chris Davis

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Being “due for one”

Khris Davis

2015

​

Batting Average: 0.247

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Home Runs: 27

24.7% of his at-bats ended in hits (rather than outs)

Source: Baseball Reference

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Khris Davis

2015 - 0.247

Source: Baseball Reference

Batting Average

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Khris Davis

2015 - 0.247

2016 - 0.247

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Source: Baseball Reference

Batting Average

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Khris Davis

2015 - 0.247

2016 - 0.247

2017 - 0.247

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Source: Baseball Reference

Batting Average

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Khris Davis

2015 - 0.247

2016 - 0.247

2017 - 0.247

2018 - 0.247

Source: Baseball Reference

Batting Average

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Khris Davis

2015 - 0.247 (97/392)

2016 - 0.247 (137/555)

2017 - 0.247 (140/566)

2018 - 0.247 (142/576)

Source: Baseball Reference

Batting Average

Hits

At-Bats

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Khris Davis

2015 - 0.247

2016 - 0.247

2017 - 0.247

2018 - 0.247

Batting Average

MLB Record for longest batting average streak*

*https://www.mlb.com/news/khris-davis-hits-247-for-4th-straight-year-c296696156

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Khris Davis

2015 - 0.247

2016 - 0.247

2017 - 0.247

2018 - 0.247

2019 – 0.220

Batting Average

A Tale of Two Khris/Chris Davis’s

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Being “due for one”

Khris Davis

2015 - 0.247

2016 - 0.247

2017 - 0.247

2018 - 0.247

2019 – 0.220

Batting Average

I guess he was “due for” a year with a different batting average! Lol jk.

A Tale of Two Khris/Chris Davis’s

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Topics

• Law of large numbers
• Simple definition of probability
• Properties of probabilities
• Probability models

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Simple definition of probability

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Simple definition of probability

This is the simple definition of probability

• Only used when all outcomes are equally likely.

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Simple definition of probability

There are 8 red marbles and 12 blue marbles in a jar. What is the probability of selecting a red marble from the jar?

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There are 8 red marbles and 12 blue marbles in a jar. What is the probability of selecting a red marble from the jar?

R: Event of selecting a red marble from the jar

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P(R): Probability of Event R occurring

Simple definition of probability

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Simple definition of probability

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Simple definition of probability

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Simple definition of probability

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Simple definition of probability

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Simple definition of probability

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Simple definition of probability

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Topics

• Law of large numbers
• Simple definition of probability
• Properties of probabilities
• Probability models

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Properties of Probabilities

Properties:

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Properties:

• Probabilities are always between 0 – 1 (0% - 100%).

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Properties of Probabilities

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Properties of Probabilities

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Properties of Probabilities

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Properties:

• Probabilities are always between 0 – 1 (0% - 100%).
• The probabilities of all possible events add to 1 (100%).

Properties of Probabilities

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Properties of Probabilities

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Properties of Probabilities

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Properties of Probabilities

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Properties of Probabilities

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Properties:

• Probabilities are always between 0 – 1 (0% - 100%).
• The probabilities of all possible events add to 1 (100%).
• Complement rule: the probability of event A not happening, P(A^C), is equal to 1 – P(A).

Properties of Probabilities

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Properties of Probabilities

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Properties of Probabilities

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Properties of Probabilities

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Properties of Probabilities

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Properties of Probabilities

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Topics

• Law of large numbers
• Simple definition of probability
• Properties of probabilities
• Probability models

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Probability Models

Probability models: List a set of outcomes and their probabilities

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Probability Models

Probability models: List a set of outcomes and their probabilities.

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Properties:

• All probabilities are between 0 – 1.
• All probabilities add to 1.

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Probability Models

I have a bag of 20 beads. They are a mix of clear/solid and red/green. There are 6 clear red beads in the bag and 8 red solid beads. There are 2 clear green beads and 4 green solid beads.

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Probability Models

I have a bag of 20 beads. They are a mix of clear/solid and red/green. There are 6 clear red beads in the bag and 8 red solid beads. There are 2 clear green beads and 4 green solid beads.

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Probability Models

1. All probabilities are between 0 – 1

2. All probabilities add to 1

I have a bag of 20 beads. They are a mix of clear/solid and red/green. There are 6 clear red beads in the bag and 8 red solid beads. There are 2 clear green beads and 4 green solid beads.

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Probability Models

1. All probabilities are between 0 – 1

2. All probabilities add to 1

I have a bag of 20 beads. They are a mix of clear/solid and red/green. There are 6 clear red beads in the bag and 8 red solid beads. There are 2 clear green beads and 4 green solid beads.

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Probability Models

1. All probabilities are between 0 – 1

2. All probabilities add to 1 🡪 0.3+0.4+0.1+0.2

I have a bag of 20 beads. They are a mix of clear/solid and red/green. There are 6 clear red beads in the bag and 8 red solid beads. There are 2 clear green beads and 4 green solid beads.

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Probability Models

1. All probabilities are between 0 – 1

2. All probabilities add to 1 🡪 0.3+0.4+0.1+0.2 = 1

I have a bag of 20 beads. They are a mix of clear/solid and red/green. There are 6 clear red beads in the bag and 8 red solid beads. There are 2 clear green beads and 4 green solid beads.

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Probability Models

Let: R = Select a red bead, Q = Select a red solid bead

1. Find P(Q): 2. Find P(R):

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3. Find P(Q^C): 4. Find P(R^C):

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Probability Models

1. Find P(Q): 2. Find P(R):

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3. Find P(Q^C): 4. Find P(R^C):

Let: R = Select a red bead, Q = Select a red solid bead

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Probability Models

1. Find P(Q): 0.40 2. Find P(R):

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3. Find P(Q^C): 4. Find P(R^C):

Let: R = Select a red bead, Q = Select a red solid bead

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Probability Models

1. Find P(Q): 0.40 2. Find P(R):

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3. Find P(Q^C): 4. Find P(R^C):

Let: R = Select a red bead, Q = Select a red solid bead

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Probability Models

1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7

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3. Find P(Q^C): 4. Find P(R^C):

Let: R = Select a red bead, Q = Select a red solid bead

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Probability Models

1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7

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3. Find P(Q^C): 4. Find P(R^C):

Let: R = Select a red bead, Q = Select a red solid bead

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Probability Models

1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7

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3. Find P(Q^C): .3+.1+.2=.6 4. Find P(R^C):

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Let: R = Select a red bead, Q = Select a red solid bead

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Probability Models

1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7

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3. Find P(Q^C): .3+.1+.2=.6 4. Find P(R^C):

Let: R = Select a red bead, Q = Select a red solid bead

0.4

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Probability Models

1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7

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3. Find P(Q^C): .3+.1+.2=.6 4. Find P(R^C):

or 1 - .4 = .6

Let: R = Select a red bead, Q = Select a red solid bead

0.4

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Probability Models

1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7

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3. Find P(Q^C): .3+.1+.2=.6 4. Find P(R^C):

or 1 - .4 = .6

Let: R = Select a red bead, Q = Select a red solid bead

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Probability Models

1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7

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3. Find P(Q^C): .3+.1+.2=.6 4. Find P(R^C): 0.1 + 0.2 = 0.3

or 1 - .4 = .6

Let: R = Select a red bead, Q = Select a red solid bead

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Probability Models

1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7

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3. Find P(Q^C): .3+.1+.2=.6 4. Find P(R^C): 0.1 + 0.2 = 0.3

or 1 - .4 = .6

0.7

Let: R = Select a red bead, Q = Select a red solid bead

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Probability Models

1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7

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3. Find P(Q^C): .3+.1+.2=.6 4. Find P(R^C): 0.1 + 0.2 = 0.3

or 1 - .4 = .6 or 1 - 0.7 = 0.3

Let: R = Select a red bead, Q = Select a red solid bead

0.7

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Lesson 8.8

Discussion

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How do police choose who to stop?

During the “Stop and Frisk” program, NYC police were allowed to stop people in public if they had a reasonable suspicion the person might soon commit a crime.

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• Ex: Stop someone wearing a trench coat during July heat.

• Allowed police to be more proactive, rather than reactive, to crime.

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*Source: https://theconversation.com/stop-and-frisk-can-work-under-careful-supervision-127785

How do police choose who to stop?

Problem: “The low burden of proof, the large discretion granted to police and the relatively invisible nature of these sorts of encounters combine to create real potential for abuse.”

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-Criminology professors Henry Fradella and Michael White*

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Problem: “The low burden of proof, the large discretion granted to police and the relatively invisible nature of these sorts of encounters combine to create real potential for abuse.”

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-Criminology professors Henry Fradella and Michael White*

Of the millions of stops, only about 12% of all stops resulted in an arrest or a summons.*

*Source: https://theconversation.com/stop-and-frisk-can-work-under-careful-supervision-127785

How do police choose who to stop?

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Are police biased in who they choose to stop?

How do police choose who to stop?

From “Stop and Frisk” protest

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Population of NYC in 2011

Source: 2011 American Community Survey (data.census.gov)

Proportion of stops in 2011 NYC sample

Note: Proportions calculated just among these three groups (the predominant groups in the dataset)

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Population of NYC in 2011

Source: 2011 American Community Survey (data.census.gov)

Proportion of stops in 2011 NYC sample

Note: Proportions calculated just among these three groups (the predominant groups in the dataset)

Discussion Question: If stops were done completely at random, which groups would be stopped proportionally more often? Less often? What conclusions can you draw from this?

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If stops were done completely at random…

Note: Proportions calculated just among these three groups (the predominant groups in the dataset)

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If stops were done completely at random…

Note: Proportions calculated just among these three groups (the predominant groups in the dataset)

Actual stops:

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If stops were done completely at random…

Note: Proportions calculated just among these three groups (the predominant groups in the dataset)

Actual stops:

Discussion Question: What conclusions can you draw from these differences?

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Lesson 8.8

Practice

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