ECON 4630ECON 5630

TOPIC #3: PROBABILITY THEORY

I.What is Probability?

  1. Definition:Probability is the relative frequency as the sample size becomes infinitely large. Alternatively, probability is the number of favorable outcomes divided by the total number of possible outcomes.
  1. Examples
  1. Objective vs. Subjective Probability

II.Probabilities of More Complex Events

  1. Probability Trees in General
  1. An Example: Gender of Children

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OutcomesProb1Prob2outcome BBBB .0625 .0731 e1 BBBG .0625 .0675 e2 BBGB .0625 .0675 e3 BBGG .0625 .0623 e4 BGBB .0625 .0675 e5 BGBG .0625 .0623 e6 BGGB .0625 .0623 e7 BGGG .0625 .0575 e8 GBBB .0625 .0675 e9 GBBG .0625 .0623 e10 GBGB .0625 .0623 e11 GBGG .0625 .0575 e12 GGBB .0625 .0623 e13 GGBG .0625 .0575 e14 GGGB .0625 .0575 e15 GGGG .0625 .0531 e16

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  1. The Special Rule of Multiplication:Assuming each outcome is independent of every other (that is, the occurrence of one outcome has no effect on the occurrence or non-occurrence of any other outcome), then
  1. Outcome Sets and Events
  2. Outcome Set

Definition: The outcome set S is the collection of all possible outcomes.

Venn Diagram:

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  1. Event

Definition: An event is a combination of outcomes. That is, an event E is a subset of S

Example:Suppose E: at least 3 girls

So E = { }

Venn Diagram:

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3.Special Rule of Addition: If outcomes are mutually exclusive, the probability of an event occurring is the sum of the probabilities of each event occurring. That is, .

Example:Suppose F: exactly 2 boys

So F = { }

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  1. Combinations of Events
  2. Union

Example: Suppose a couple would be sorry if they had fewer than 2 boys OR if all 4 kids were of the same gender.

Let:

G: fewer than 2 boys

J: all same gender

Venn Diagram:

G = { }

J = { }

= { }

P() =

  1. Intersection

Example: Suppose a couple would be sorry if they had fewer than 2 boys AND if all 4 kids were of the same gender.

Let:

G: fewer than 2 boys

J: all same gender

Venn Diagram:

G = { }

J = { }

= { }

P() =

  1. The General Rule of Addition:

Recall the Special Rule of Addition:

  1. Complement

Definition: The complement of E, , is all the points that are not in E.

  1. Conditional Probability

Definition: Conditional probability is the probability of some event occurring given that some other event has occurred.

Notation:

Example: Suppose we know that G (fewer than 2 boys) has occurred. Given this, what is the probability that J (all same gender) will occur?

The General Rule of Multiplication:

  1. Review Example

Suppose a restaurant finds that 75% of all customers use chili sauce, 80% use salt, and 65% use both.

  1. What is the probability that a particular customer uses at least one of these two condiments?
  1. What is the probability that a salt user uses chili sauce? Equivalently, what is the probability that a customer will use chili sauce given that he or she uses salt?
  1. Independence

Definition: If the occurrence of event A is unaffected by the occurrence or non-occurrence of event B, A and B are independent of each other.

Note: Generally speaking, independence must be proved mathematically.

Method of Proving Independence #1:

Method of Proving Independence #2:

Example #1: Suppose we gather data on the performance of students in ECON 4630 and which hand each student writes with. Suppose our survey results are as follows:

Region / Left-Handed / Right Handed
Good: A or B / 0.100 / 0.700
Less Good: C and below / 0.025 / 0.175

Is one’s performance independent of which hand one writes with?

Example #2: Suppose we ask Americans whether or not they believe Jon Gosselinis to blame for the breakup of the family depicted in the TV series Jon and Kate plus 8. Suppose our survey results are as follows:

Respondent’s Gender / JG to blame / JG not to blame
Male / 0.078 / 0.250
Female / 0.621 / 0.051

Is one’s opinion on this matter independent of one’s gender?

Example #3:

A personnel officer for a firm that employs many part-time salespersons tries out a new sales aptitude test on several hundred applicants. Because the test is unproven, results are not used in hiring. 40% of all applicants show high aptitude on the test and 12% of those hired show both high aptitude and achieve good sales records. The firm’s experience is that 30% of all salespersons achieve good sales.

Let A be the event “shows high aptitude.”

Let B be the event “achieves good sales.”

  1. Find P(A), P(B), P(AB), and P(BA).
  1. Are A and B independent? Prove this mathematically.
  1. Combinations of Random Variables
  2. Definition: A random variable is a real-valued set function whose value is determined by the outcome of an experiment.
  1. Examples:
  1. Notation:
  1. Probability Distributions in General

Outcome / Probability / outcomes
P(X=0)
P(X=1)
P(X=2)
P(X=3)
P(X=4)
  1. Calculating the mean:

X / P(x) / xP(x)
0 / 0.0731
1 / 0.2700
2 / 0.3738
3 / 0.2300
4 / 0.0531
  1. Calculating the variance:

X / P(x) / (x-) / (x-)2 / (x-)2P(x) / x2P(x)
0 / 0.0731
1 / 0.2700
2 / 0.3738
3 / 0.2300
4 / 0.0531
  1. Rules for Transforming Random Variables
  1. Example: Let X = number of dots that turn up on a die

x / P(x) / xP(x) / (x-x)2P(x) / x2P(x)
1
2
3
4
5
6

Suppose this game has a payoff that is a linear function of X:, Specifically, suppose Y = 2X + 8.

y / P(y) / yP(y) / (y-x)2P(y) / y2P(y)
  1. Joint Probability Distributions

Definition: Joint probability is the probability that two or more events will occur at the same time.

Example 1: Consider rolling a pair of dice. Let X = the number of threes and Y = the number of fives.

There are 36 possible outcomes:

1st die / 2nd die / X / Y
1 / 1 / 0 / 0
1 / 2 / 0 / 0
1 / 3 / 1 / 0
1 / 4 / 0 / 0
1 / 5 / 0 / 1
1 / 6 / 0 / 0
2 / 1 / 0 / 0
2 / 2 / 0 / 0
2 / 3 / 1 / 0
2 / 4 / 0 / 0
2 / 5 / 0 / 1
2 / 6 / 0 / 0
3 / 1 / 1 / 0
3 / 2 / 1 / 0
3 / 3 / 2 / 0
3 / 4 / 1 / 0
3 / 5 / 1 / 1
3 / 6 / 1 / 0
4 / 1 / 0 / 0
4 / 2 / 0 / 0
4 / 3 / 1 / 0
4 / 4 / 0 / 0
4 / 5 / 0 / 1
4 / 6 / 0 / 0
5 / 1 / 0 / 1
5 / 2 / 0 / 1
5 / 3 / 1 / 1
5 / 4 / 0 / 1
5 / 5 / 0 / 2
5 / 6 / 0 / 1
6 / 1 / 0 / 0
6 / 2 / 0 / 0
6 / 3 / 1 / 0
6 / 4 / 0 / 0
6 / 5 / 0 / 1
6 / 6 / 0 / 0
/ 0 / 1 / 2
0
1
2

Example 2: Suppose we survey Denton residents regarding their satisfaction with restaurant choices in Denton. Let X measure satisfaction with restaurant choices (with 4 being very satisfied) and Y measure length of residency (1 = 5 or fewer years, 2 = 6 or more). Perhaps our survey comes up with the following:

/ 1 / 2 / 3 / 4
1 / 0.04 / 0.14 / 0.23 / 0.07
2 / 0.07 / 0.17 / 0.23 / 0.05
  1. Independence
  1. Conditional Probability
  1. Covariance: how variables vary together

Method of calculating covariance #1:

Method of calculating covariance #2:

Example: Diameter and Usable Height of Trees (in feet)

/ 20 / 25 / P(d)
1 / 0.16 / 0.09 / 0.25
1.25 / 0.15 / 0.30 / 0.45
1.5 / 0.03 / 0.17 / 0.20
1.75 / 0.00 / 0.10 / 0.10
P(h) / 0.34 / 0.66 / 1.00

Marginal Distributions:

Mean and Variance of D and H:

d / H / P(d,h) / /
1 / 20 / 0.16
1 / 25 / 0.09
1.25 / 20 / 0.15
1.25 / 25 / 0.30
1.5 / 20 / 0.03
1.5 / 25 / 0.17
1.75 / 20 / 0.00
1.75 / 25 / 0.10

What does covariance tell us?

What does covariance not tell us?

  1. Correlation

Example: Diameter and Usable Height of Trees (in inches)

/ 240 / 300 / P(d)
12 / 0.16 / 0.09 / 0.25
15 / 0.15 / 0.30 / 0.45
18 / 0.03 / 0.17 / 0.20
21 / 0.00 / 0.10 / 0.10
P(h) / 0.34 / 0.66 / 1.00

Marginal Distributions:

Mean and Variance of D and H:

D / H / P(d,h) / /
12 / 240 / 0.16
12 / 300 / 0.09
15 / 240 / 0.15
15 / 300 / 0.30
18 / 240 / 0.03
18 / 300 / 0.17
21 / 240 / 0.00
21 / 300 / 0.10

Correlation coefficient:

NOTES:

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