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
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Topics
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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)
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?
Situation 1: HTTHHTHTHT
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?
Situation 1: HTTHHTHTHT
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
Batting Average: 0.247
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
Batting Average: 0.247
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
Batting Average: 0.262
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
Batting Average: 0.262
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
Batting Average: 0.262
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
Batting Average:
Home Runs:
Source: Baseball Reference
A Tale of Two Khris/Chris Davis’s
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Being “due for one”
Chris Davis
2018
Batting Average: 0.168
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
Batting Average: 0.168
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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
skewthescript.org
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:
“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
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
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
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*
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
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Simple definition of probability
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Simple definition of probability
This is the simple definition of probability
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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
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
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Properties of Probabilities
Properties:
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Properties:
Properties of Probabilities
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Properties of Probabilities
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Properties of Probabilities
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Properties:
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:
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
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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.
Properties:
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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
Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
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.
Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
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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.
Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
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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.
Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
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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.
Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
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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):
3. Find P(QC): 4. Find P(RC):
Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
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Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
Probability Models
1. Find P(Q): 2. Find P(R):
3. Find P(QC): 4. Find P(RC):
Let: R = Select a red bead, Q = Select a red solid bead
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Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
Probability Models
1. Find P(Q): 0.40 2. Find P(R):
3. Find P(QC): 4. Find P(RC):
Let: R = Select a red bead, Q = Select a red solid bead
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Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
Probability Models
1. Find P(Q): 0.40 2. Find P(R):
3. Find P(QC): 4. Find P(RC):
Let: R = Select a red bead, Q = Select a red solid bead
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Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
Probability Models
1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7
3. Find P(QC): 4. Find P(RC):
Let: R = Select a red bead, Q = Select a red solid bead
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Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
Probability Models
1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7
3. Find P(QC): 4. Find P(RC):
Let: R = Select a red bead, Q = Select a red solid bead
skewthescript.org
Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
Probability Models
1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7
3. Find P(QC): .3+.1+.2=.6 4. Find P(RC):
Let: R = Select a red bead, Q = Select a red solid bead
skewthescript.org
Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
Probability Models
1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7
3. Find P(QC): .3+.1+.2=.6 4. Find P(RC):
Let: R = Select a red bead, Q = Select a red solid bead
0.4
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Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
Probability Models
1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7
3. Find P(QC): .3+.1+.2=.6 4. Find P(RC):
or 1 - .4 = .6
Let: R = Select a red bead, Q = Select a red solid bead
0.4
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Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
Probability Models
1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7
3. Find P(QC): .3+.1+.2=.6 4. Find P(RC):
or 1 - .4 = .6
Let: R = Select a red bead, Q = Select a red solid bead
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Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
Probability Models
1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7
3. Find P(QC): .3+.1+.2=.6 4. Find P(RC): 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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Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
Probability Models
1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7
3. Find P(QC): .3+.1+.2=.6 4. Find P(RC): 0.1 + 0.2 = 0.3
or 1 - .4 = .6
0.7
Let: R = Select a red bead, Q = Select a red solid bead
skewthescript.org
Bead Type | clear red | red solid | clear green | green solid |
Probability | | | | |
Probability Models
1. Find P(Q): 0.40 2. Find P(R): 0.3 + 0.4 = 0.7
3. Find P(QC): .3+.1+.2=.6 4. Find P(RC): 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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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.”
-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.”
-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.*
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
Race-Ethnicity | Black | Hispanic | White |
Population | 2,054,101 | 2,373,304 | 2,731,173 |
Source: 2011 American Community Survey (data.census.gov)
Proportion of stops in 2011 NYC sample
Race-Ethnicity | Black | Hispanic | White |
Proportion of Stops | 55.1% | 35.2% | 9.7% |
Note: Proportions calculated just among these three groups (the predominant groups in the dataset)
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Population of NYC in 2011
Race-Ethnicity | Black | Hispanic | White |
Population | 2,054,101 | 2,373,304 | 2,731,173 |
Source: 2011 American Community Survey (data.census.gov)
Proportion of stops in 2011 NYC sample
Race-Ethnicity | Black | Hispanic | White |
Proportion of Stops | 55.1% | 35.2% | 9.7% |
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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Race-Ethnicity | Black | Hispanic | White |
Population | 2,054,101 | 2,373,304 | 2,731,173 |
Proportion of Stops | | | |
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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Race-Ethnicity | Black | Hispanic | White |
Population | 2,054,101 | 2,373,304 | 2,731,173 |
Proportion of Stops | 2,054,101 7,158,578 =28.6% | 2,373,304 7,158,578 =33.2% | 2,731,173 7,158,578 =38.2% |
If stops were done completely at random…
Note: Proportions calculated just among these three groups (the predominant groups in the dataset)
Race-Ethnicity | Black | Hispanic | White |
Proportion of Stops | 55.1% | 35.2% | 9.7% |
Actual stops:
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Race-Ethnicity | Black | Hispanic | White |
Population | 2,054,101 | 2,373,304 | 2,731,173 |
Proportion of Stops | 2,054,101 7,158,578 =28.6% | 2,373,304 7,158,578 =33.2% | 2,731,173 7,158,578 =38.2% |
If stops were done completely at random…
Note: Proportions calculated just among these three groups (the predominant groups in the dataset)
Race-Ethnicity | Black | Hispanic | White |
Proportion of Stops | 55.1% | 35.2% | 9.7% |
Actual stops:
Discussion Question: What conclusions can you draw from these differences?
skewthescript.org
Lesson 8.8
Practice
skewthescript.org