In Chapters 1-3, We Saw How to Analyze Things As They Actually Are

Levine Chapter 4

Basic Probability

Introduction

Place in the Course

In Chapters 1-3, we saw how to analyze things as they actually are.

In this chapter, we begin looking at how to analyze things that are only probable.

The first task, of course, is to learn how to think about probability.

Probability

We represent probability by P.
If something is completely certain, P=1.0
If something is absolutely not going to happen, P = 0.0
The real world is always somewhere in between. /

Types of Probability

A priori / Probability is estimated by reasoning. / The probability of pulling an ace out of a full deck of cards is 4/52 (1/13)
Empirical / Probability is estimated from data. / Based on recent experience, the probability of a visitor buying something if they visit our site is 0.15 (15%)
Subjective / Probability is based on personal experience and guesswork. / I’m 80% certain that that our store traffic will be down today because it is raining.

Events and Sample States

Events

Each possible outcome of a variable is an event.

A simple event is described by a single characteristic. / What is the probability that a household will purchase a big-screen television in the next year?
A joint event is an even that has two or more characteristics. / What is the probability that a household will buy both a big screen TV and a VCR?
The complement of an event A (represented by the symbol A’) includes all events that are not part of A. / What is the probability that a family will not purchase a big screen TV next year?

Sample Space

The collection of all possible events is a sample space.

From a sample of 1,000 households:

Planned to
Purchase / Actually Purchased
Yes / No / Total
Yes / 200 / 50 / 250
No / 100 / 650 / 750
Total / 300 / 700 / 1000

Simple Event

A simple event is described by a single characteristic.

The probability that a family planned to purchase a big screen.

= 250 / 1000 = .25 (25%)

Planned to
Purchase / Actually Purchased
Yes / No / Total
Yes / 200 / 50 / 250
No / 100 / 650 / 750
Total / 300 / 700 / 1000

What is the probability that a family actually purchased a big screen TV?

Joint Event

A joint event is an even that has two or more characteristics.

The probability that a family planned to purchase a big-screen.

= 250 / 1000 = .25 (25%)

Planned to
Purchase / Actually Purchased
Yes / No / Total
Yes / 200 / 50 / 250
No / 100 / 650 / 750
Total / 300 / 700 / 1000

What is the probability that a family said that they planned to purchase a big-screen TV but actually did not?

Complement of an Event

The complement of an event A (represented by the symbol A’) includes all events that are not part of A.

The probability that a family did not plan to purchase a big screen.

= 750 / 1000 = .75 (75%)

Planned to
Purchase / Actually Purchased
Yes / No / Total
Yes / 200 / 50 / 250
No / 100 / 650 / 750
Total / 300 / 700 / 1000

What is the probability that a family did not plan to purchase a big screen TV and did not purchase it?

Review Question

U.S. Tax Code / Income Level
Less than $50,000 / $50,000 or More / Total
Fair / 225 / 180 / 405
Unfair / 280 / 320 / 600
Total / 505 / 500 / 1,005

1.Give an example of a simple event.

2.Give an example of a joint event

3.What is the complement of “tax code is fair?”

4.In what category would you put someone earning $50,000 per year?

5.What proportion of the respondents believe that the tax code is fair?

6.What proportion of the respondents make $50,000 or more?

7.What is the proportion of people who make over $50,000 believe that the tax code is unfair?

8.What proportion of people making under $50,000 believe that the tax code is fair?

Contingency Tables and VENN Diagrams

Venn Diagrams

Two characteristics

(A) Planned to purchase

(B) Did purchase

Intersect (Upside-down U): Both characteristics are satisfied.

Example: Planned to purchase AND did purchase.

Union (U): At least one characteristic is satisfied.

Either planned to purchase OR did purchase.

Contingency Table
Planned to Purchase / Actually Purchased
Yes / No / Total
Yes / 200 / 50 / 250
No / 100 / 650 / 750
Total / 300 / 700 / 1000
/ Venn Diagram:

Simple Probability

P (Actually Purchased) = 200 / 300

P (Planned to Purchase) = ?

Joint Probability

P (Actually Purchased AND Planned to Purchase) = 200 / 1000

P (Planned to Purchase AND Did NOT Purchase) = ?

Marginal Probability

Planned to
Purchase / Actually Purchased
Yes / No / Total
Yes / 200 / 50 / 250
No / 100 / 650 / 750
Total / 300 / 700 / 1000

Marginal probability is a set of joint probabilities

Planned to Purchase =
Planned to Purchase AND purchased +
Planned to Purchase and did not Purchase.

= 200/1000 + 50/1000 = 250/1000 = 0.25

Events must be mutually exclusive and collectively exhaustive.

General Addition Rule

P (A OR B) = P(A) + P(B) – P(A AND B)

Planned to
Purchase / Actually Purchased
Yes / No / Total
Yes / 200 / 50 / 250
No / 100 / 650 / 750
Total / 300 / 700 / 1000

Probability of Planned to purchase OR Actually Purchased =

Probability of Planned to Purchase

+ Did Purchase

- Probability of Both Planned and Did

= 250/1000 + 300/1000 – 200/1000

= (250+300-200) / 1000 = 350/1000

Conditional Probability

Sometimes, with additional information, you can refine your probability estimate.

Computing Conditional Probabilities

Probability of A given B is P(A|B)

P(A|B) = P(A AND B) / P(B)

P(A AND B) is the joint probability of A and B

P(B) is the marginal probability of B

Planned to
Purchase / Actually Purchased
Yes / No / Total
Yes / 200 / 50 / 250 / 2. But if you know that the customer planned to purchase (and therefore in this row), the probability that they will make a purchase is 200/250 = .8 (80%)
No / 100 / 650 / 750
Total / 300 / 700 / 1000 / 1. Without additional information, the probability that a customer will make a purchase is 300/1000 = .3 (30%)

Example

P(A) Without additional information, the probability a new customer will make a purchase is

= 300 / 1000 = 0.3

But suppose you know that the person said that they planned to make a purchase.

P(A|B) Did Purchase given Planned to Purchase =

= P (A AND B) Both Planned to Purchase and Did Purchase = 200

/ P(B) Planned to Purchase 250

= 0.8

So you may be able to make a better estimate if you have more information.

Decision Trees

Contingency Table

Planned to
Purchase / Actually Purchased
Yes / No / Total
Yes / 200 / 50 / 250
No / 100 / 650 / 750
Total / 300 / 700 / 1000

Decision Tree

Summary problem

U.S. Tax Code / Income Level
Less than $50,000 / $50,000 or More / Total
Fair / 225 / 180 / 405
Unfair / 280 / 320 / 600
Total / 505 / 500 / 1,005

1.What proportion of the respondents believe that the tax code is fair?

2.What is the proportion of people who make over $50,000 believe that the tax code is fair?

3.What proportion of people making under $50,000 believe that the tax code is fair?

4.You know that someone makes $27,000 per year. How likely are they to believe that the tax code is unfair?

5.You know that someone believes that the tax code is fair. How likely are they to make over $50,000?