Probability Models - Ds...

Probability Models

Section 6.2

The Language of Probability

What is random?

Empirical means that it is based on observation rather than theorizing.

Probability describes what happens in MANY trials.

Example 6.9: Long-term relative frequency

Randomness and Probability We call a phenomenon random if

individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.

The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, probability is long-term relative frequency.

Assignment

Page 410 exercises 6.21 – 6.28

Toss a coin…

We cannot know the outcome in advance.

The outcome will be either heads or tails.

Each of these outcomes has the probability of ½.

The basis of all probability models is a list of all possible outcomes and a probability for each outcome.

Sample Spaces

The sample space S of a random phenomenon is the set of all possible outcomes.

To specify S, we must state what constitutes an individual outcome and then state which outcomes can occur.

How to count!

Being able to properly enumerate the outcomes in a sample space will be critical to determining probabilities.

Two techniques are very helpful in making sure you don’t accidentally overlook any outcomes.

These techniques are the tree diagram and the multiplication principle.

Tree Diagram

Toss

a coin

H

T

Roll a

die

1

2

3

4

5

6

1

2

3

4

5

6

H1

H2

H3

H4

H5

H6

T1

T2

T3

T4

T5

T6

Multiplication Principle

If you can do one task in n number of ways and a second task in m number of ways, then both tasks can be done in nXm number of ways.

Nondiscrete sample space

Some sample spaces are simply too large to allow all of the possible outcomes to be listed.

Generate a random decimal number between 0 and 1.

– Nondiscrete sample spaces

With and Without Replacement

Sampling with replacement means that once you’ve made your first selection, you return it so that it can be chosen again.

Sampling without replacement means that you do not return your first selection.

Assignment

Page 416, problems 6.29 – 6.36

Probability of an Event

The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes.

P(A) = The Number Of Ways Event A Can Occur

The Total Number Of Possible Outcomes

A pair of dice is rolled, one black and one white. Find the probability of each of the following events.

1. The total is 10.

2. The total is at least 10.

3. The total is less than 10.

4. The total is at most 10.

5. The total is 7.

6. The total is 2.

7. The total is 13.

8. The numbers are 2 and 5.

9. The black die has 2 and the white die has 5.

10.The black die has 2 or the white die has 5.

112

16

56

1112

16

136

0

118

136

13

Probability Rules All probabilities are between 0 and 1 inclusive

The sum of all the probabilities in the sample space is 1

The probability of an event which cannot occur is 0.

The probability of any event which is not in the sample space is zero.

The probability of an event which must occur is 1.

The probability of an event not occurring is one minus the probability of it occurring.

P(E') = 1 - P(E)

0 ( ) 1P E

The Addition Rule

Two events are disjoint (mutually exclusive) if they have no outcomes in common.

If two events are disjoint, the number of ways one or the other can occur is

( or ) ( ) ( )n A B n A n B

Set Notation

Union

Empty Event

Intersect

Examples

6.13 Complement Rule

6.14 Applying Probability Rules

6.15 Applying Probability Rules

6.16 Applying Probability Rules

Assignment

Page 423, problems 6.37 – 6.44

Independence and the multiplication rule

Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs.

If A and B are independent,

( and ) ( ) ( )n A B n A n B

Independent or not independent?

Example 6.17

Example 6.18

Applying the Multiplication Rule

Example 6.19

Independence and the Complement Rule

Example 6.21

Assignment

Page 430, exercises 6.45 – 6.52

Section Exercises

Page 432, exercises 6.53 – 6.63