Counting Techniques and Basic Probability Modelscathy/Math2311/Lectures/chapter...Counting...

Counting Techniques and Basic Probability ModelsSections 2.2 & 2.3

Cathy Poliak, Ph.D.cathy@math.uh.edu

Office hours: T Th 2:30 pm - 5:15 pm 620 PGH

Department of MathematicsUniversity of Houston

February 2, 2016

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 1 / 32

Outline

1 Popper

2 Counting Techniques

3 Sets

4 Venn Diagrams

5 Introduction Probability

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 2 / 32

Popper Set Up

Fill in all of the proper bubbles for you ID number.

Use a #2 pencil.

This is popper number 01.

When you fill in the bubbles make sure it is dark.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 3 / 32

Counting Rules

If an experiment can be described as a sequence of k steps withn1 possible outcomes on the first step, n2 possible outcomes onthe second step, and so on, then the total number of experimentaloutcomes is given by (n1)(n2) . . . (nk ).Permutations: allows one to compute the number of outcomeswhen r objects are to be selected from a set of n objects wherethe order of selection is important. The number of permutations isgiven by Pn

r = n!(n−r)!

When we allow repeated values, The number of orderings of nobjects taken r at a time, with repetition is nr .The number of permutations, P, of n objects taken n at a time withr objects alike, s of another kind alike, and t of another kind alikeis P = n!

r !s!t!

The number of circular permutations of n objects is (n − 1)!.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 4 / 32

Combinations

Counts the number of experimental outcomes when the experimentinvolves selecting r objects from a (usually larger) set of n objects. Thenumber of combinations of n objects taken r unordered at a time is

Cnr =

(nr

)=

n!r !(n − r)!

Rcode: choose(n,r)

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 6 / 32

Examples

In how many ways can a committee of 5 be chosen from a groupof 12 people?

In a manufacturing company they have to choose 5 out of 50boxes to be sent to a store. How many ways can they choose the5 boxes?

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 7 / 32

Definitions

A set is a collection of objects.

The items that are in a set called elements.

We typically denote a set by capital letters of the English alphabet.

Examples: A = {knife, spoon, fork}, B = {2,4,6,8}.

The set B could also be written asB = {x |x are even whole numbers between 0 and 10}.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 8 / 32

Notations of Sets

Notation Descriptiona ∈ A The object a is an element of the set A.A ⊆ B Set A is a subset of set B.

That is every element in A is also in B.A ⊂ B Set A is a proper subset of set B.

That is every element that is is in A is also in set B andthere is at least one element in set B that is no in set A.

A ∪ B A set of all elements that are in A or B.A ∩ B A set of all elements that are in A and B.U Called the universal set, all elements we are interested in.AC The set of all elements that are in the universal set

but are not in set A.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 9 / 32

Examples

The following are sets: U = {1,2,3,4,5,6,7,8,9,10},A = {1,2,3,4,5,6,9,10}, B = {3,4,7,8}, and C = {2,3,9,10}

C ⊂ AA ∪ B = {1,2,3,4,5,6,7,8,9,10} = U9 ∈ A but 9 /∈ BAC = {7,8}A ∩ B = {3,4}AC ∩ C = ∅ These means that the sets AC and C are disjoint.(B ∪ C)C = {1,5,6}A ∩ B ∩ C = {3}

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 10 / 32

Definitions

A Venn diagram is a very useful tool for showing the relationshipsbetween sets.

Venn diagrams consist of a rectangle with one or more shapes(usually circles) inside the rectangle.

The rectangle represents all of the elements that we areinterested in for a given situation. This set is the universal set.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 12 / 32

Graph of Venn Diagrams

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 13 / 32

Graph of Disjoint Events

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 14 / 32

A ∩ B

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 15 / 32

A ∪ B

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 16 / 32

AC ∩ B

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 17 / 32

A ∩ B ∩ C

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 18 / 32

Soft Drink Preference

A group of 100 people are asked about their preference for soft drinks.The results are as follows: 55 like Coke, 25 like Diet Coke, 45 likePepsi, 15 like Coke and Diet Coke, 5 like all 3 soft drinks, 25 like Cokeand Pepsi, 5 only like Diet Coke (nothing else). Fill in the the Venndiagram with these numbers.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 19 / 32

Winning the State Lottery

Suppose a person won the Jackpot of the State Lottery five timesin a row.

What do you think would happen?

Is it possible for a person to win the state lottery five consecutivetimes?

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 21 / 32

Randomness and Probability

We call a phenomenon random if individual outcomes areuncertain.

However, there is a regular distribution of outcomes in a largenumber of repetitions.

Chance behaviors unpredictable in the short run but has a regularand predictable pattern in the long run.

Long-run must be infinitely long to give them frequencies ofenough time to even out.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 22 / 32

Proportion of Heads in Long Run

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Trial A proportions

Trial B Proprotions

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1

13

25

37

49

61

73

85

97

10

9

12

1

13

3

14

5

15

7

16

9

18

1

19

3

Trial A proportions

Trial B Proprotions

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 23 / 32

Probability

Probability is a value between zero and one, inclusive.

If probability is zero this indicates that the event will never occur.

If probability is one this indicates that the event will always occur.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 24 / 32

Why Study Probability in Statistics?

Statistical Idea: Rare event rule for inferential statistics

If, under a given assumption, the probability of a particularobserved event is extremely small, we conclude that theassumption is probably not correct.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 25 / 32

Some Definitions About Probability

The sample space of a random phenomenon is the set of allpossible outcomes. S is used to denote sample space.

An event is an outcome or a set of outcomes of a randomphenomenon. An event is a subset of the sample space.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 26 / 32

Examples of Sample Space

Random Outcomes Sample SpacePhenomenon

Two live Record the sequence of the S = {BB,GB,BG,GG}births births being either

boy or girlTwo live Count the S = {0,1,2}births number of girlsFill a Determine the S = {X ≥ 0}

can of soda volume in ouncesFlip a Count the number S = {0,1,2,3}

coin 3 times of heads up

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 27 / 32

Assigning probabilities

Classical method is use when all the experimental outcomes areequally likely. If n experimental outcomes are possible, aprobability of 1/n is assigned to each experimental outcome.Example: Drawing a card from a standard deck of 52 cards. Eachcard has a 1/52 probability of being selected.Relative frequency method is used when assigning probabilitiesis appropriate when data are available to estimate the proportionof the time the experimental outcome will occur if the experimentis repeated a large number of times. That is for any event E ,probability of E is

P(E) =number of times E occurs

total number of observations

=n(E)

n(S)

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 28 / 32

Example of Probabilities

Relative frequency method: An insurance company determined thenumber of accidents in a year. A sample of 100 people were surveyedto determine the number of accidents they were in a year: 0 accidents25 people, 1 accident 45 people, 2 accidents 20 people, 3 or moreaccidents 10 people. The following table shows the relative frequencyfor the outcomes.

Number of accidents Frequency (count) Relative frequency

0 25 25100 = 0.25

1 45 45100 = 0.45

2 20 20100 = 0.20

3 or more 10 10100 = 0.10

Total 100 1Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 29 / 32

Example 2

If 5 marbles are drawn at random all at once from a bag containing 8white and 6 black marbles, what is the probability the 2 will be whiteand 3 will be black?

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 30 / 32

Example 3

The qualified applicant pool for six management trainee positionsconsists of seven women and five men.

1. What is the probability that a randomly selected trainee class willconsist entirely of women?

2. What is the probability that a randomly selected trainee class willconsist of an equal number of men and women?

Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH (Department of Mathematics University of Houston )Sections 2.2 & 2.3 February 2, 2016 31 / 32