Probability

I Introduction to Probability

A Satisfactory outcomes vs. total outcomes

B Basic Properties

C Terminology

II Combinatory Probability

A The Addition Rule – “Or”

1. The special addition rule (mutually exclusive events)

2. The general addition rule (non-mutually exclusive events)

B The Multiplication Rule – “And”

1. The special multiplication rule (for independent events)

2. The general multiplication rule (for non-independent events)

Probability for Equally Likely Outcomes

Suppose an experiment has N possible outcomes, all equally likely. Then the probability that a specified event occurs equals the number of ways, f, that the event can occur, divided by the total number of possible outcomes. In symbols

Probability of a given event = N

f

Number of ways a given event can occur

Total of all possible outcomes

Frequency distribution of annual income for U.S. families

Probability from Frequency Distributions

What is the a priori probability of having an income between $15,000 and $24,999

Frequency distribution for students’ ages

N = 40

Frequency distribution for students’ ages

What is the likelihood of randomly selecting a student who is older than 20 but less than 22?

What is the likelihood of selecting a student who’s age is an odd number?

What is the likelihood of selecting a student who is either 21 or 23?

Sample space for rolling a die once

Possible outcomes for rolling a pair of dice

Probabilities of 2 throws of the die

• What is the probability of a 1 and a 3?

• What is the probability of two sixes?

• What is the probability of at least one 3?

2/36

1/36

12/36

The Sum of Two Die TossesSum Frequency2 13 24 35 46 57 68 59 410 311 212 1

What is the probability that the sum will be

5?

7?

What is the probability that the sum will be 10 or more?

What is the probability that the sum will be either 3 or less or 11 or more?

4/36

6/36

6/36

3/36 + 3/36

Two computer simulations of tossing a balanced coin 100 times

Basic Properties of Probabilities

Property 1: The probability of an event is always between 0 and 1, inclusive.

Property 2: The probability of an event that cannot occur is 0. (An event that cannot occur is called an impossible event.)

Property 3: The probability of an event that must occur is 1. (An event that must occur is called a certain event.)

A deck of playing cards

The event the king of hearts is selected

1/52

The event a king is selected

1/13 = 4/52

The event a heart is selected

1/4 = 13/52

The event a face card is selected

3/13=13/52

Sample Space and Events

Sample space: The collection of all possible outcomes for an experiment.

Event: A collection of outcomes for the experiment, that is, any subset of the sample space.

Probability Notation

If E is an event, then P(E) stands for the probability that event E occurs. It is read “the probability of E”

Venn diagram for event E

Relationships Among Events

(not E): The event that “E does not occur.”

(A & B): The event that “both A and B occur.”

(A or B): The event that “either A or B or both occur.”

Event (not E) where E is the probability of drawing a face card.

40/52=10/13

An event and its complement

The Complementation Rule

For any event E,

P(E) = 1 – P (~ E).

In words, the probability that an event occurs equals 1 minus the probability that it does not occur.

Combinations of Events

The Addition Rule – “Or”

• The special addition rule (mutually exclusive events)

• The general addition rule (non-mutually exclusive events)

The Multiplication Rule – “And”

• The special multiplication rule (for independent events)

• The general multiplication rule (for non-independent events)

Venn diagrams for (a) event (not E )(b) event (A & B) (c) event (A or B)

Event (B & C)

1/13 X 1/4 = 1/52

Event (B or C)

16/52 = 4/52 + 13/52-1/52

Event (C & D)

3/52 = 3/13 X 1/4

Mutually Exclusive Events

Two or more events are said to be mutually exclusive if at most one of them can occur when the experiment is performed, that is, if no two of them have outcomes in common

Two mutually exclusive events

(a) Two mutually exclusive events(b) Two non-mutually exclusive events

(a) Three mutually exclusive events (b) Three non-mutually exclusive events (c) Three non-mutually exclusive events

The Special Addition Rule

If event A and event B are mutually exclusive, then

More generally, if events A, B, C, … are mutually exclusive, then

That is, for mutually exclusive events, the probability that at least one of the events occurs is equal to the sum of the individual probabilities.

BPAPBAP or

... ... or or CPBPAPCBAP

Non-mutually exclusive events

The General Addition Rule

If A and B are any two events, then

P(A or B) = P(A) + P(B) – P(A & B).

In words, for any two events, the probability that one or the other occurs equals the sum of the individual probabilities less the probability that both occur.

P(A or B): Spade or Face Card

P (spade) + P (face card) – P (spade & face card) = 1/4 + 3/13 – 3/52= 22/52

The Special Multiplication Rule (for independent events)

If events A, B, C, . . . are independent, then

P(A & B & C & ) = P(A) P(B)P(C)

What is the probability of all of these events occurring:

1. Flip a coin and get a head

2. Draw a card and get an ace

3. Throw a die and get a 1

P(A & B & C ) = P(A) · P(B) · P(C) = 1/2 X 1/13 X 1/6

Conditional Probability: For non-independent events

The probability that event B occurs given that event A has occurred is called a conditional probability. It is denoted by the symbol P(B | A), which is read “the probability of B given A.” We call A the given event.

Contingency Table for Joint Probabilities

Contingency table for age and rank of faculty members (using frequencies)

The Conditional-Probability Rule

If A and B are any two events, then

In words, for any two events, the conditional probability that one event occurs given that the other event has occurred equals the joint probability of the two events divided by the probability of the given event.

.)(

)&()|(

AP

BAPABP

.)(

)&()|(

AP

BAPABP

The Conditional-Probability Rule

P( R3 | A4 ) =

= 36/253

= 0.142

P( A4 | R3 ) =

= 36/320

= 0.112

Joint probability distribution (using proportions)

.)(

)&()|(

AP

BAPABP

P( R3 | A4 ) =

= 0.031/0.217

= 0.142

P( A4 | R3 ) =

= 0.031/.0275

= 0.112

Contingency table of marital status and sex(using proportions)

.)(

)&()|(

AP

BAPABP Joint probability

distribution (using proportions)

The General Multiplication Rule

If A and B are any two events, then

P(A & B) = P(A) P(B | A).

In words, for any two events, their joint probability equals the probability that one of the events occurs times the conditional probability of the other event given that event.

Note: Either 1) The events are independent and then P(A & B) = P(A) · P(B).Or2) The events are not independent and then a contingency table must be used

Independent Events

Event B is said to be independent of event A if the occurrence of event A does not affect the probability that event B occurs. In symbols,

P(B | A) = P(B).

This means that knowing whether event A has occurred provides no probabilistic information about the occurrence of event B.

Class Fr So Ju Se

Male 40 50 50 40 | 180

Female 80 100 100 80 | 360

120 150 150 120 | 540

Probability and the Normal Distribution

• What is the probability of randomly selecting an individual with an I.Q. between 95 and 115? Mean 100, S.D. 15.

• Find the z-score for 95 and 115 and compute the area between

More Preview of Experimental Design Using probability to evaluate a treatment effect. Values that are extremely

unlikely to be obtained from the original population are viewed as evidence of a treatment effect.

A Preview of Sampling Distributions

X X

• What is the probability of randomly selecting a sample of three individuals, all of whom have an I.Q. of 135 or more?

• Find the z-score of 135, compute the tail region and raise it to the 3rd power.

• This concept is critical to understanding future concepts

So while the odds chance selection of a single person this far above the mean is not all that unlikely, the odds of a sample this far above the mean are astronomical

z = 2.19 P = 0.0143 0.01433 = 0.0000029

Summary

For multiple events there are two rules:“AND” (multiplication) and “OR” (addition)

There are just a few special considerations:1. For the “And” rule, if the events are not

independent, you don’t multiply, you use a table.2. For the “Or” rule, if the events are not mutually

exclusive you have to subtract off their double count