Lecture 5 (Sept 16, 2015)

Last Day

More graphical displays

Histograms

Cumulative frequency plots

Pareto diagrams

Digidot, time series and scatter plots

Today

Review of graphical displays

Introduction to probability

Basic definitions

Tree diagrams

Venn diagrams

Stem (year)

Leaf (tenth of a year) Frequency

1 8 1

2 3 8 2

3 0 2 3 7 7 9 6

4 0 1 1 2 3 3 3 4 5 5 6 7 8 8 9 9 16

5 0 1 2 4 5 6 6 8 8

6 1 3 2

7 0

8 4 1

Basic Probability

Probability of success = number of successful outcomes/total number of outcomes

Experiment: any process that generates a set of data Ex: tossing a coin to see how many times you find

heads out of 100 tosses

Ex: study effect of different feedstocks on biogas production in an anaerobic digester

Observation: any recording of information, whether numerical (continuous) or categorical (discrete)

Basic Probability

Sample space: set of all possible outcomes of a statistical experiment Each outcome in a sample space is called an

element or member of the sample space, or sample point

If a sample space has a finite number of elements, you can list the members in enclosed brackets Ex: Members of sample space (S) for the

tossing of a coin, with H = heads and T = tails

S = {H, T}

Sample Space

Ex: Consider the experiment of the tossing of a die. What is the sample space, S?

S = {1, 2, 3, 4, 5, 6}

Ex: What is the sample space if we are only interested in whether the top of the die is even or odd?

S = {even, odd}

Sample Space

For sample spaces with a very large number of data points, we define them by a statement or rule instead of listing the potential sample points in brackets

Ex: if all possible outcomes in a sample space are the cities in the world with a population greater than 500,000, we can write

S = {x x is a city with a population > 500,000}

Reads: S is the set of all x such that x is a city with a population > 500,000

Tree Diagrams

A tree diagram is a graphical means to list potential outcomes of an experiment

Useful to help determine more complex sample spaces

Example 5-1: Tree Diagram 1

An experiment consists of flipping a coin, then flipping it again if heads occurs. If tails occurs, then a die is tossed once. Draw the tree diagram for this sample space and list the possible outcomes.

Example 5-2: Tree Diagram 2

Three items are selected at random from a manufacturing process. Each item is inspected and classified as defective, D, or not defective, N. Draw the tree diagram for this sample space and list all possible outcomes.

Events

For any given experiment, we may be interested in the occurrence of a certain event, which represents a subset of the sample space

Ex: we might be interested in event A that the outcome when a die is tossed is evenly divisible by 3

S = {1,2,3,4,5,6} all possible outcomes

A = {3,6} event that the outcome is evenly divisible by 3

Events

An event is just data with some similar characteristic that constitutes a subset of the sample space It is possible that the event may include the

entire sample space S An event might contain no sample points or

elements This is called a null set and given by

The complement (A) of an event (A) with respect to sample space (S) is the subset of all elements in S that are not in A.

S = {1,2,3,4,5,6} A = {3,6} A = {1,2,4,5}

Venn Diagrams

Venn diagrams are used to help visualize events and sample spaces

3

6

1

2

4

5

A

A

Rectangle encloses the entire sample

space

Circle represents subset A within larger sample

space. The circle encloses some portion of the

data that have a similar

characteristic. The data outside the

circle do not have this characteristic

Represents complement of

subset A

Intersection of Events

Suppose C and D are two events associated with the sample space S S = {1,2,3,4,5,6}

C is the event that an even number comes up C = {2,4,6}

D is the event that a number greater than 3 comes up

D = {4,5,6}

Intersection of Events

The subset that represents the intersection of C and D is

C D = {4,6}

The intersection of two events C and D (given by C D) contains only

elements that are common to C and D

Venn Diagram (Intersection of Events)

2

4

6

5

1

3

C

D

Sample space: rolling a die

Event C: top number

is even

Event D: top number is > 3

C D Intersection of events given by area of intersecting circles on Venn diagram

Multiple Events

For some events, say A and B, we might be interested in the situation where either A or B occurthis is the union of A and B The union of two events A and B is

denoted A B and contains all the elements that belong to A or B or both

Ex: A = {a,b,c}

B = {b,c,d,e}

then A B = {a,b,c,d,e}

Multiple Events

2

4

6

5 1

3

A

B

C

A B = {1,2,3,4,5}

What is A, B, and C complements?

Mutually Exclusive Events

It is possible that two events, say E and F, cannot both occur simultaneously. The events E and F are said to be mutually exclusive.

Two events E and F are mutually exclusive, or disjoint, if E F =

E and F have no elements in common

Venn Diagram (Mutually Exclusive Events)

1

E F

E F =

2

E F = {1,2} and

intersection or

union

Additional Sample Space Concepts

A = intersection with a null set is a null set

A = A union of an event and a null set is the event A A = intersection of an event and its complement is the null set

A A = S union of an event and its complement is the sample space S = complement of the sample space is a null set

(A B) = A B complement of the union of A and B is the intersection of the complement of A and the complement of B

(A B) = A B complement of the intersection of A and B is the union of the complement of A and the complement of B

(A B) = A + B - A B = addition rule A B = A x B multiplication rule (A U B) C = (A C) U (B C) and (A B) U C = (A U C) (B U C)=distributive law

Next day

More probability Venn diagram examples

Re-learning to count

This material is covered in Chapter 2 in the text