Probability ModelsTarget Goals I can define

and list a sample space for an event.I can use

basic probability rules.

- Weve done models for lines and for distributions

of data (normal density curves). - Now we are going to give a mathematical

description or model for randomness.

Sample Space

- The set of all possible outcomes of an event is

the sample space S of the event. - Example For the event roll a die and
- observe what number it lands on. The
- sample space contains all possible
- numbers the die could land on.
- S 1, 2, 3, 4, 5, 6

An Event

- An event is an outcome (or a set of outcomes)

from a sample space. - Example 1 When flipping three coins,
- an event may be getting the result
- HTH.
- In this case, the event is one
- outcome from the sample space.

- Example 2 When flipping three coins,
- an event may be getting two tails.
- In this case, the event is a set of outcomes

(HTT, TTH, THT) from the sample space. - An event is usually denoted by a capital letter.
- For example, call getting two tails
- The probability of event A is denoted

event A.

P(A).

- Example Roll the Dice
- Give a probability model for the chance process

of rolling two fair, six-sided dice one thats

red and one thats green.

- Probability Rules

Since the dice are fair, each outcome is equally

likely. Each outcome has probability 1/36.

Roll a 5 is an event that we will call A.

- List the possible outcomes for A
- A (1, 4),(2, 3),(3, 2),(4,1)
- What is the P(A)?
- P(A) 4/36 or 11.11

- To specify S, we must state what constitutes an

individual outcome and then state which outcomes

can occur. - You must be able to list the outcomes in a

sample space.

Listing Outcomes

- Example Random Digit
- If you let your pencil fall on the Table of

random digits, what are the possible outcomes S? - S 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Real Life

- In statistics, what might tossing a coin

represent? - An opinion poll with a yes or no answer.
- Any two outcome event.

Probability Rules

- These facts follow from the idea of probability

as the long-run proportion of repetitions on

which an event occurs.

Rule 1

- The probability of any event is between 0 and 1.
- Probability of 0 indicates the event will never

occur. - Probability of 1 indicates the event will always

occur. - 0 P (A) 1

Rule 2

- If S is the sample space, then P(S) 1.
- All possible outcomes in the sample space add up

to 1.

Rule 3

- The probability that event A does not occur is

one minus the probability that A does occur. - Called the complement of A
- Is denoted Ac.
- P(Ac ) 1 P(A)

These complementary events can be shown on a Venn

Diagram. E 2, 4, 6 and EC 1, 3, 5

Let the circle represent event E.

Let the rectangle represent the sample space.

Let the shaded area represent event not E.

- Example
- When flipping two coins, the probability of

getting two heads is - (.5)(.5) 0.25.
- The probability of not getting two heads is

1 0.25 0.75.

Rule 4

- If events A and B are disjoint then,
- They have no outcomes in common.
- The events never occur simultaneously.
- The probability that A or B occurs is the

probability that A occurs plus the probability

that B occurs. - P(A or B) P(A) P(B)

A Venn Diagram for the roll of a six-sided die

and the following two events A 2 B

6

A and B are mutually exclusive (disjoint) since

they have no outcomes in common

The intersection of A and B is empty!

3

4

6

2

1

5

Example

- Let event A be rolling a die and landing on an

even number, and - event B be rolling a die and landing on an odd

number. - The outcomes for A are 2, 4, 6 and the outcomes

for B are 1, 3, 5. - These events are disjoint because they have no

outcomes in common.

- So the probability of A or B (landing on either

an even or an odd number) equals the probability

of A plus the probability of B. - P(A) P(B) ½ ½ 1

Disjoint Events

(mutually exclusive)

- Ways to express
- A U B s/a A or B, or
- A n B empty set

Venn diagram

- Venn diagram showing disjoint (mutually

exclusive) events A and B.

Complement

- A U Ac S
- A n Ac empty set
- Venn diagram showing the complement Ac of an

event A.

Example Marital Status of Young Women

- Draw a woman age 25 to 34 years old at random

and record her marital status. - What does at random mean?
- Every woman has the same chance to be chosen.

Here is the probability model

Marital status Never married Married Widowed Divorced

Probability 0.298 0.622 0.005 0.075

- Why do the probabilities add up to 1?
- P(S) 1

- Find the probability the woman drawn is not

married by the complement rule. - P(not married) 1 P(married)
- P(not married) 1 0.622 0.378

Are never married and divorced are disjoint

events?

- Yes, because a woman cant be both never

married and divorced at the same time. - P(never married or divorced)
- P(never married) P(divorced)
- 0.298 0.075 0.373

Example Benfords Law

- Fake numbers in tax returns and other settings

often display patterns that arent present in

legitimate records. - The first digits in legitimate records often

follow a distribution called Benfords Law.

Benfords probability of first digits. Note

0 not a leading digit here.

First digit 1 2 3 4 5 6 7 8 9

Probability .301 .176 .125 .097 .079 .067 .058 .051 .046

- a. Events A first digit 1,
- B first digit is 6 or greater
- P(A)
- P(B)

P(1) .301

P(6) P(7) P(8) P(9)

0.222

Describe in words P(Ac).

- A first digit 1
- First digit anything but a 1
- P(Ac) 1 - P(A)
- 1 - 0.301
- 0.699

b. Are events A and B disjoint?

- A first digit 1,
- B first digit is 6 or greater
- Yes, they cant occur together so
- P(A or B) P(A) P(B)
- 0.301 0.222
- 0.523

c. P(C) first odd digit

- P(1) P(3) P(5) P(7) P(9)
- 0.609

d. Is P(B or C) disjoint?

- B first digit is 6 or greater
- P(C) first odd digit
- No, we cant add the probabilities.
- Outcomes 7 and 9 are common to both events so,
- P(1) P(3) P(5) P(6) P(7) P(8) P(9)
- 0.7227

Equally Likely Outcomes

- Most random phenomena do not have equally likely

outcomes. - When they do, the probability of any event A is
- P(A) count of outcomes in A
- count of outcomes in S

Find the P(B) using equally likely outcomes.

- Event B first digit is 6 or greater
- P(B) P(6) P(7) P(8) P(9)
- 1/9 1/9 1/9 1/9 4/9
- 0.444
- Summary
- A crook who tries to use random digits will end

up with too many first digits 6 or greater.

vs. Benfords 0.222