ge 210 lecture 5 (descriptive stats and intro to probability)
DESCRIPTION
probability and staticsTRANSCRIPT
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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
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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
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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)
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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}
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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}
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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
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Tree Diagrams
A tree diagram is a graphical means to list potential outcomes of an experiment
Useful to help determine more complex sample spaces
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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.
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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.
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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
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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}
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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
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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}
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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
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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
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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}
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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?
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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
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Venn Diagram (Mutually Exclusive Events)
1
E F
E F =
2
E F = {1,2} and
intersection or
union
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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
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Next day
More probability Venn diagram examples
Re-learning to count
This material is covered in Chapter 2 in the text