The Multiplication Rule & Conditional Probability

- Lesson 8.10 -

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A 36-year-old single, heterosexual man wonders…

How can I make my online profile more appealing?

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Insidious social standards favor:

Tall

Wealthy

Heterosexual

White Dudes

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A 36-year-old single, heterosexual man wonders…

How can I make my online profile more appealing?

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A 36-year-old single, heterosexual man wonders…

• Tall
• Wealthy

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A 36-year-old single, heterosexual man wonders…

• Tall
• Wealthy

Today’s Key Analysis

Can we use probability to find the secure men (the “honest” profiles)?

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Lesson 8.10

Guided Notes

Handout: skewthescript.org/5-3

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Topics

• Conditional probability
• Independence
• The multiplication rule
• Tree diagrams
• 5 Intuitive Probability Rules for a 5

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Topics

• Conditional probability
• Independence
• The multiplication rule
• Tree diagrams
• 5 Intuitive Probability Rules for a 5

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The Data

• Public online dating profiles from 2012 on OKCupid.
• People who identified as age 36, male, and heterosexual, living in the San Francisco area.
• Collected data for each profile that reported their height and yearly earnings.

Full data source: https://github.com/wetchler/okcupid

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The Data

Why are we looking at just the age 36 profiles?

• It’s a lower-bound for the age when men typically earn their median salaries.
• It’s an age when men who are dating online might really start feeling the pressure to find someone.

For pay data by age and gender: https://www.payscale.com/data/peak-earnings

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The Data

• Estimated 2012 median yearly earnings for individual men in San Francisco county: $59,397

​

• Male median height (U.S.): 69.2 in. (5’9”)

​

1. Earnings median is an upper-bound estimate based on median earnings for workers in San Francisco county per the 2012 American Community Survey (data.census.gov).

2. Height median is from the CDC: National Health Statistics Reports, “Mean Body Weight, Height, Waist Circumference, and Body Mass Index Among Adults: United States, 1999–2000 Through 2015–2016.” Center for Disease Control and Prevention, 2018.

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Addition rule in probability models

We’ll categorize the 192 men in the OKCupid dataset using the following convention:

Socially valued

Socially de-valued

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Conditional Probability

Condition: a “given” in a problem

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Conditional Probability

Condition: a “given” in a problem

​

P(A|B) = Probability of A occurring given that B has already occurred.

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101

48

22

21

Conditional Probability

In the OkCupid data, we find 21 short low-earners, 22 tall low-earners, 48 short high-earners, and 101 tall high-earners.

​

T = event of selecting a tall man

​

H = event of selecting a high-income earner

​

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101

48

22

21

Conditional Probability

1. P(T) =

​

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101

48

22

21

Conditional Probability

1. P(T) =

​

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101

48

22

21

Conditional Probability

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101

48

22

21

Conditional Probability

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101

48

22

21

Conditional Probability

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101

48

22

21

Conditional Probability

Total # of possibilities

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101

48

22

21

Conditional Probability

2. P(T|H) =

​

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101

48

22

21

Conditional Probability

2. P(T|H) =

​

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101

48

22

21

Conditional Probability

New # of possibilities

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101

48

22

21

Conditional Probability

New # of possibilities

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101

48

22

21

Conditional Probability

New # of possibilities

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101

48

22

21

Conditional Probability Formula

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101

48

22

21

Conditional Probability Formula

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101

48

22

21

Conditional Probability Intuition

P(T|H) =

​

​

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Conditional Probability Intuition

101

48

22

21

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Conditional Probability Intuition

101

48

22

21

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Conditional Probability Intuition

101

48

22

21

“Given” means divide by the given

​

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Conditional Probability Intuition: “Given” means divide by the given.

Conditional Probability

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Probability formulas can be confusing.

​

My advice: if possible, use intuitive rules and reason instead of formulas.

Conditional Probability

Conditional Probability Intuition: “Given” means divide by the given.

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Conditions in two-way tables

1. P(T) =

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Conditions in two-way tables

1. P(T) =

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Conditions in two-way tables

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Conditions in two-way tables

2. P(T|H^C) =

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Conditions in two-way tables

2. P(T|H^C) =

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Conditions in two-way tables

2. P(T|H^C) =

“Given” means divide by the given

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Conditions in two-way tables

“Given” means divide by the given

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Conditions in two-way tables

“Given” means divide by the given

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Conditions in two-way tables

“Given” means divide by the given

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Two-way probability tables

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Conditions in two-way tables

Raw Counts

Percentages

2. P(T|H^C)

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Conditions in two-way tables

Raw Counts

Percentages

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Conditions in two-way tables

Raw Counts

Percentages

“Given” means divide by the given

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Topics

• Conditional probability
• Independence
• The multiplication rule
• Tree diagrams
• 5 Intuitive Probability Rules for a 5

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Independence

Two events (A & B) are independent if knowing the outcome of one event does not affect the probability that the other event will occur.

​

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Independence

Two events (A & B) are independent if knowing the outcome of one event does not affect the probability that the other event will occur.

​

P(A|B) = P(A)

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Independence

1. P(T) = 64.1%

Probability that they’re tall, without knowing about their earnings.

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Independence

1. P(T) = 64.1%

2. P(T|H^C) = 51.2%

Probability that they’re tall, when we know they are a low-earner.

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Independence

1. P(T) = 64.1%

2. P(T|H^C) = 51.2%

When we knew the person was a low-earner (when H^C was “given”), the probability of selecting a tall person shrunk. So, the events of selecting a low earner and someone who is tall are not independent.

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Independence

1. P(T) = 64.1%

2. P(T|H^C) = 51.2%

In other words, their earnings gave us information about their height. So earnings and height are not independent.

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Is income really associated with height?

Numerous studies have found that taller people tend to earn more $$$.

​

Note:

Correlation ≠ Causation

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1. P(T) = 64.1%

2. P(T|H^C) = 51.2%

Does income/height correlation explain this dependence in the OKCupid data?

​

Discussion Preview

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1. P(T) = 64.1%

2. P(T|H^C) = 51.2%

Does income/height correlation explain this dependence in the OKCupid data?

​

Not necessarily

🡪 We’ll explore more in the discussion.

Discussion Preview

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Independent vs. Mutually Exclusive

Mutually exclusive: when events that have no intersection (i.e. they cannot both occur)

A

B

0.80

0.20

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Independent vs. Mutually Exclusive

Mutually exclusive: when events that have no intersection (i.e. they cannot both occur)

A

B

0.80

0.20

P(A) = 0.80

P(A|B) =

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Independent vs. Mutually Exclusive

Mutually exclusive: when events that have no intersection (i.e. they cannot both occur)

A

B

0.80

0.20

P(A) = 0.80

P(A|B) =

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Independent vs. Mutually Exclusive

Mutually exclusive: when events that have no intersection (i.e. they cannot both occur)

A

B

0.80

0.20

P(A) = 0.80

P(A|B) = 0.00

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Independent vs. Mutually Exclusive

Mutually exclusive: when events that have no intersection (i.e. they cannot both occur)

A

B

0.80

0.20

P(A) = 0.80

P(A|B) = 0.00

Mutually exclusive events are not independent. Knowing that one event occurs greatly affects the probability of the other event (lowers it to 0).

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Topics

• Conditional probability
• Independence
• The multiplication rule
• Tree diagrams
• 5 Intuitive Probability Rules for a 5

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Formal Multiplication Rule

The formal multiplication rule (all events)…

For independent events only…

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Formal Multiplication Rule

The formal multiplication rule (all events)…

For independent events only…

Independence:

P(B) = P(B|A)

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Intuitive Multiplication Rule

• “And” means multiply
• Account for dependent events

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Topics

• Conditional probability
• Independence
• The multiplication rule
• Tree diagrams
• 5 Intuitive Probability Rules for a 5

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Tree Diagram

Models multiple dependent or successive events (events that depend on each other or that happen right after each other).

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Tree Diagram

Models multiple dependent or successive events (events that depend on each other or that happen right after each other).

Each branch represents a different event

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Tree Diagram (Dream School)

1. You have a 0.65 probability of getting higher than average GPA and ACT scores.

2. If you have a higher than average GPA/ACT, you have a 0.83 chance of being admitted.

3. If you have a below average GPA/ACT score, you have a 0.39 chance of being admitted.

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Event 1: GPA/ACT Scores

Tree Diagram (Dream School)

1. You have a 0.65 probability of getting higher than average GPA and ACT scores.

2. If you have a higher than average GPA/ACT, you have a 0.83 chance of being admitted.

3. If you have a below average GPA/ACT score, you have a 0.39 chance of being admitted.

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Event 1: GPA/ACT Scores

Higher than average ACT/GPA

Below average ACT/GPA

Tree Diagram (Dream School)

1. You have a 0.65 probability of getting higher than average GPA and ACT scores.

2. If you have a higher than average GPA/ACT, you have a 0.83 chance of being admitted.

3. If you have a below average GPA/ACT score, you have a 0.39 chance of being admitted.

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Event 1: GPA/ACT Scores

Below average ACT/GPA

0.65

0.35

Higher than average ACT/GPA

Tree Diagram (Dream School)

1. You have a 0.65 probability of getting higher than average GPA and ACT scores.

2. If you have a higher than average GPA/ACT, you have a 0.83 chance of being admitted.

3. If you have a below average GPA/ACT score, you have a 0.39 chance of being admitted.

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Event 1: GPA/ACT Scores

Below average ACT/GPA

0.65

0.35

Event 2: Admission to College

Higher than average ACT/GPA

Tree Diagram (Dream School)

1. You have a 0.65 probability of getting higher than average GPA and ACT scores.

2. If you have a higher than average GPA/ACT, you have a 0.83 chance of being admitted.

3. If you have a below average GPA/ACT score, you have a 0.39 chance of being admitted.

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Below average ACT/GPA

0.65

0.35

Higher than average ACT/GPA

Tree Diagram (Dream School)

1. You have a 0.65 probability of getting higher than average GPA and ACT scores.

2. If you have a higher than average GPA/ACT, you have a 0.83 chance of being admitted.

3. If you have a below average GPA/ACT score, you have a 0.39 chance of being admitted.

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Below average ACT/GPA

0.65

0.35

Admitted to school

Not admitted to school

Higher than average ACT/GPA

Tree Diagram (Dream School)

1. You have a 0.65 probability of getting higher than average GPA and ACT scores.

2. If you have a higher than average GPA/ACT, you have a 0.83 chance of being admitted.

3. If you have a below average GPA/ACT score, you have a 0.39 chance of being admitted.

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Below average ACT/GPA

0.65

0.35

Admitted to school

Not admitted to school

0.83

Higher than average ACT/GPA

Tree Diagram (Dream School)

1. You have a 0.65 probability of getting higher than average GPA and ACT scores.

2. If you have a higher than average GPA/ACT, you have a 0.83 chance of being admitted.

3. If you have a below average GPA/ACT score, you have a 0.39 chance of being admitted.

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Below average ACT/GPA

0.65

0.35

Admitted to school

Not admitted to school

0.83

0.17

Higher than average ACT/GPA

Tree Diagram (Dream School)

1. You have a 0.65 probability of getting higher than average GPA and ACT scores.

2. If you have a higher than average GPA/ACT, you have a 0.83 chance of being admitted.

3. If you have a below average GPA/ACT score, you have a 0.39 chance of being admitted.

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Below average ACT/GPA

0.65

0.35

Admitted to school

Not admitted to school

0.83

0.17

Admitted to school

Not admitted to school

Higher than average ACT/GPA

Tree Diagram (Dream School)

1. You have a 0.65 probability of getting higher than average GPA and ACT scores.

2. If you have a higher than average GPA/ACT, you have a 0.83 chance of being admitted.

3. If you have a below average GPA/ACT score, you have a 0.39 chance of being admitted.

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Below average ACT/GPA

0.65

0.35

Admitted to school

Not admitted to school

0.83

0.17

Admitted to school

Not admitted to school

0.39

Higher than average ACT/GPA

Tree Diagram (Dream School)

1. You have a 0.65 probability of getting higher than average GPA and ACT scores.

2. If you have a higher than average GPA/ACT, you have a 0.83 chance of being admitted.

3. If you have a below average GPA/ACT score, you have a 0.39 chance of being admitted.

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Below average ACT/GPA

0.65

0.35

Admitted to school

Not admitted to school

0.83

0.17

Admitted to school

Not admitted to school

0.39

0.61

Higher than average ACT/GPA

Tree Diagram (Dream School)

1. You have a 0.65 probability of getting higher than average GPA and ACT scores.

2. If you have a higher than average GPA/ACT, you have a 0.83 chance of being admitted.

3. If you have a below average GPA/ACT score, you have a 0.39 chance of being admitted.

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Find:

1. P(H∩A) =

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Find:

1. P(H∩A) =

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

0.65

Find:

1. P(H∩A) =

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

0.65

“and” means multiply!

Find:

1. P(H∩A) =

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

0.65

“and” means multiply!

Find:

1. P(H∩A) =

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

0.65

Find:

1. P(H∩A) = 0.54

“and” means multiply!

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Find:

2. P(A) =

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Find:

2. P(A) =

0.65

Path 1

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Find:

2. P(A) =

0.65

Path 1

Or

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Find:

2. P(A) =

0.65

Path 2

Path 1

Or

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Find:

2. P(A) =

0.65

Path 2

Path 1

Or

“or” means add!

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Find:

2. P(A) =

0.65

Path 2

Path 1

Or

“or” means add!

+

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Find:

2. P(A) = 0.68

0.65

Path 2

Path 1

Or

“or” means add!

+

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Find:

3. P(H|A) =

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Find:

3. P(H|A) =

Below average ACT/GPA

0.35

Admitted to school

Not admitted to school

0.83

0.17

Admitted to school

Not admitted to school

0.39

0.61

Higher than average ACT/GPA

0.65

Path 2

Path 1

+

P(A) = 0.68

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Find:

3. P(H|A) =

Below average ACT/GPA

0.35

Admitted to school

Not admitted to school

0.83

0.17

Admitted to school

Not admitted to school

0.39

0.61

Higher than average ACT/GPA

0.65

Path 2

Path 1

+

P(A) = 0.68

“given” means divide by the given!

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Below average ACT/GPA

0.35

Admitted to school

Not admitted to school

0.83

0.17

Admitted to school

Not admitted to school

0.39

0.61

Higher than average ACT/GPA

0.65

Path 2

Path 1

+

P(A) = 0.68

“given” means divide by the given!

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Below average ACT/GPA

0.35

Admitted to school

Not admitted to school

0.83

0.17

Admitted to school

Not admitted to school

0.39

0.61

Higher than average ACT/GPA

0.65

Path 2

Path 1

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Below average ACT/GPA

0.35

Admitted to school

Not admitted to school

0.83

0.17

Admitted to school

Not admitted to school

0.39

0.61

Higher than average ACT/GPA

0.65

Path 2

Path 1

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Below average ACT/GPA

0.35

Admitted to school

Not admitted to school

0.83

0.17

Admitted to school

Not admitted to school

0.39

0.61

Higher than average ACT/GPA

0.65

Path 2

Path 1

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Tree Diagram

Let:

H = Event of getting higher than average ACT/GPA

​

A = Event of being admitted to your dream school

Find:

3. P(H|A) = 0.79

Below average ACT/GPA

0.35

Admitted to school

Not admitted to school

0.83

0.17

Admitted to school

Not admitted to school

0.39

0.61

Higher than average ACT/GPA

0.65

Path 2

Path 1

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Topics

• Conditional probability
• Independence
• The multiplication rule
• Tree diagrams
• 5 Intuitive Probability Rules for a 5

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5 Intuitive Probability Rules for a 5

​

• Probabilities are between 0 – 1 (inclusive)
• Complement rule: P(A^C) = 1 – P(A)
• “Or” means add, beware of double-counts
• “Given” means divide by the given
• “And” means multiply, adjust for dependence

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5 Intuitive Probability Rules for a 5

​

• Probabilities are between 0 – 1 (inclusive)
• Complement rule: P(A^C) = 1 – P(A)
• “Or” means add, beware of double-counts
• “Given” means divide by the given
• “And” means multiply, adjust for dependence

Use these rules and your intuition. Don’t depend on formulas for probability!

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5 Intuitive Probability Rules for a 5

​

• Probabilities are between 0 – 1 (inclusive)
• Complement rule: P(A^C) = 1 – P(A)
• “Or” means add, beware of double-counts
• “Given” means divide by the given
• “And” means multiply, adjust for dependence

Use these rules and your intuition. Don’t depend on formulas for probability!

skewthescript.org

5 Intuitive Probability Rules for a 5

​

• Probabilities are between 0 – 1 (inclusive)
• Complement rule: P(A^C) = 1 – P(A)
• “Or” means add, beware of double-counts
• “Given” means divide by the given
• “And” means multiply, adjust for dependence

Use these rules and your intuition. Don’t depend on formulas for probability!

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Lesson 8.10

Discussion

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OKCupid Profiles

If we randomly sample among the OKCupid profiles of age 36 heterosexual men…

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OKCupid Profiles

If we randomly sample among the OKCupid profiles of age 36 heterosexual men…

P(above median height):

​

P(above median income):

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OKCupid Profiles

If we randomly sample among the OKCupid profiles of age 36 heterosexual men…

P(above median height): 64.1%

​

P(above median income):

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OKCupid Profiles

If we randomly sample among the OKCupid profiles of age 36 heterosexual men…

P(above median height): 64.1%

​

P(above median income): 77.6%

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OKCupid Profiles

If we randomly sample among the OKCupid profiles of age 36 heterosexual men…

P(above median height): 64.1%

​

P(above median income): 77.6%

These probabilities are only 50% in the population!

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OKCupid Profiles

If we randomly sample among the OKCupid profiles of age 36 heterosexual men…

P(above median height): 64.1%

​

P(above median income): 77.6%

We have particular reasons to doubt the earnings are accurate…

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OKCupid Profiles

But what if we just look at the people who reported that they make below median earnings?

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Low-Earner = Honesty?

1. P(T) = 64.1%

Probability that they’re tall, without knowing about their earnings.

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1. P(T) = 64.1%

2. P(T|H^C) = 51.2%

Probability that they’re tall, when we know they are a low-earner.

Low-Earner = Honesty?

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If we find the conditional distribution (“given” low-earner), we’re close to the 50-50 split we’d expect around the median height!

Low-Earner = Honesty?

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If we find the conditional distribution (“given” low-earner), we’re close to the 50-50 split we’d expect around the median height!

Low-Earner = Honesty?

Discussion Question: Do you believe that filtering results for just the men who reported below-median earnings would return more honest matches? Explain your reasoning.

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