probability and counting techniquescathy/math3339/lecture/lecture2_3339_online.pdf1 introduction to...

Probability and Counting Techniques3.1 - 3.4

Cathy Poliak, [email protected]

Department of MathematicsUniversity of Houston

Lecture 2 - online

Cathy Poliak, Ph.D. [email protected] (Department of Mathematics University of Houston )3.1 - 3.4 Lecture 2 - online 1 / 43

Outline

1 Introduction to Probability

2 Sets

3 Venn Diagrams

4 Probability

5 Counting Techniques

6 Probability

7 Probability Rules


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?


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.


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


Probability

"A probability is a numerical value that measures the likelihoodthat an uncertain event occurs." Business Statistics:Communicating with Numbers, Jaggia and Kelly, pg 96

The probability of any outcome of a random phenomenon is theproportion of times the outcome would occur in a very long seriesof repetitions.


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.


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.Usually, Ei

Examples: E1 = knife, spoon, fork, E2 = 2,4,6,8.

The set E2 could also be written asE2 = x |x are even whole numbers between 0 and 10.

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


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,this is the union

A ∩ B A set of all elements that are in A and B,this is the intersection

Ω Called the universal set, all elements we are interested in.∼A The set of all elements that are in the universal set

but are not in set A.other notations Ac , A’, A⋃

i Ei E1 ∪ E2 ∪ . . ., the union of multiple sets⋂i Ei E1 ∩ E2 ∩ . . ., the intersection of multiple sets


Examples

The following are sets: Ω = 1,2,3,4,5,6,7,8,9,10,E1 = 1,2,3,4,5,6,9,10, E2 = 3,4,7,8, and E3 = 2,3,9,10


Examples



More Examples


1. What elements are in ∼E2 ∩∼ E3?

2. What elements are in E2 ∩∼ E1?


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.


Graph of Venn Diagrams


Graph of Disjoint Events


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.


Probability Models

An probability measure is a function which assign numbersbetween 0 and 1 to any event in the sample space Ω.

If the sample space Ω, the collection of events, and the probabilitymeasure are all specified, they constitute a probability model ofthe random experiment.


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 outcome, A,probability of A is

P(E) =number of times E occurs

total number of observations=

#(E)

N

P(E) is a probability model for any event E that is a subset of Ω.


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. [email protected] (Department of Mathematics University of Houston )3.1 - 3.4 Lecture 2 - online 19 / 43

Pair of Dice

What is the probability of getting a sum of 5?


How to get #(E)

#(E) denotes the number of elements in the subset E .

Sometimes it is not a obvious as the previous example. Thus weneed to use some counting techniques to determine #(E).


Beginning Example

In the city of Milford, applications for zoning changes go through atwo-step process:

1. A review by the panning commission.2. A final decision by the city council.

At step 1 the planning commission reviews the zoning changerequest and makes a positive or negative recommendationconcerning the change.At step 2 the city council reviews the planning commission’srecommendation and then votes to approve or to disapprove thezoning change.

How many possible decisions can be made for a zoning change inMilford?


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 ).A tree diagram can be used as a graphical representation invisualizing a multiple-step experiment.


Tree diagram

disapprove

disapprove

approve

positive

Step 1

Planning Commission

negative

Step 2

City Council

approve

Sample Points

(positive, approve)

(positive, disapprove)

(negative, approve)

(negative, disapprove)


Examples

1. How many ways can we select 4 digits and 3 letters?I If digits and letters are allowed to repeat?

I If digits are letters are not allowed to repeat?

2. In how many ways can 4 people be seated in 6 seats?


Allowing Repeated Values

When we allow repeated values, The number of orderings of n objectstaken r at a time, with repetition is nr .

Example 3: In how many ways can you write 4 letters on a tagusing each of the letters C O U G A R with repetition?


Permutations

It allows one to compute the number of outcomes when r objects areto be selected from a set of n objects where the order of selection isimportant. The number of permutations is given by

Pnr =

n!

(n − r)!

Where n! = n(n − 1)(n − 2) · · · (2)(1)

Rocode for n!: factorial(n)


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)


Examples

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

5. 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?


Difference Between Combinations and Permutations


Example 6

From a committee of 10 people.a) In how many ways can we choose a chair person, a vice-chair

person, and a secretary, assuming that one person cannot holdmore than one position?

b) In how many ways can we select a subcommittee of 3 people?


Example

7. A researcher randomly selects 3 fish from a tank of 12 and putseach of the 3 fish into different containers. How many ways canthis be done?

8. Among 10 electrical components 2 are known not to function. If 5components are randomly selected, how many ways can we haveonly one of components not functioning?


Assigning Probability

Suppose there are N total possible outcomes, the probabilityassigned to each is 1

N .

Consider an event A, with N(A) denoting the number of outcomescontained in A. Then,

P(A) =N(A)

N

Example 9: When two dice are rolled separately, there are N = 36outcomes. We want to know what is the probability of the event A= sum of two numbers = 7.


Example 10

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


Example 11

Suppose a box contains 3 defective light bulbs and 12 good bulbs.Suppose we draw a simple random sample of 4 light bulbs.

1. Find the probability that one of the bulbs drawn is defective.

2. Find the probability that half of the bulbs drawn are defective.


Example 11 Continued

3. What is the probability that none of bulbs drawn are defective?

4. What is the probability that at least one of the bulbs drawn isdefective?


Example 12

Suppose we select randomly 4 marbles drawn from a bag containing 8white and 6 black marbles. What is the probability that at least 2 of themarbles drawn are white?


Basic Probability Rules

1. 0 ≤ P(E) ≤ 1 for each event E .

2. P(Ω) = 1

3. If E1,E2, . . . is a finite or infinite sequence of events such thatEi ∩ Ej = ∅ for i 6= j , then P(

⋂i Ei) =

∑i P(Ei). If Ei ∩ Ej = ∅ for

all i 6= j we say that the events E1,E2, . . . are pairwise disjoint.


Other Probability Rules

4. Complement Rule: P(E ∩∼ F ) = P(E)− P(E ∩ F ). In particular,P(∼E) = 1− P(E).

5. P(∅) = 0

6. Addition Rule: P(E ∪ F ) = P(E) + P(F )− P(E ∩ F ).

7. If E1 ⊆ E2 ⊆ . . . is an infinite sequence, thenP(⋃

i Ei) = limi→∞P(Ei).

8. IF E1 ⊇ E2 ⊇ . . . is an infinite sequence, thenP(⋂

i Ei) = limi→∞P(Ei).


Examples of Probability Rules

13. Consider randomly selecting a student at a certain university, andlet A denote the event that the selected individual has a Visacredit card and B be the analogous event for a MasterCard.Suppose that P(A) = 0.5, P(B) = 0.4, and P(A ∩ B) = 0.25.a) Compute the probability that the selected individual has at least one

of the two types of cards.

b) What is the probability that the selected individual has neither type ofcard?

c) Describe, in terms of A and B, the event that the selected studenthas a Visa card but not a MasterCard, and the calculate theprobability of this event.


Example 14

A computer4 consulting firm presently has bids out on three projects.Let Ai = awarded project i , for i = 1,2,3, and suppose thatP(A1) = 0.22, P(A2) = 0.25, P(A3) = 0.28,P(A1 ∩ A2) = 0.11,P(A1 ∩ A3) = 0.05, P(A2 ∩ A3) = 0.07, and P(A1 ∩ A2 ∩ A3) = 0.01.Express in words each of the following events, and compute theprobability for each event.


a) A1 ∪ A2

b) ∼ A1∩ ∼ A2

c) A1 ∪ A2 ∪ A3

d) ∼ A1∩ ∼ A2∩ ∼ A3

e) ∼ A1∩ ∼ A2 ∩ A3

f) (∼ A1∩ ∼ A2) ∪ A3


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