basic probability modelscathy/math2311/lectures/spring 18/lecture5.pdf · suppose a box contains 3...

Basic Probability ModelsSection 2.3 & 2.4

Cathy Poliak, Ph.D.cathy@math.uh.eduOffice: Fleming 11c

Department of MathematicsUniversity of Houston

lecture 5 - 2311

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 1 / 27

Outline

1 Assigning Probabilities

2 Basic Rules of Probability

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 2 / 27

Popper Set Up

Fill in all of the proper bubbles.

Make sure your ID number is correct.

Make sure the filled in circles are very dark.

This is popper number 01.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 3 / 27

Popper 01 Questions

The following Venn diagram is based on 100 students that whereasked if they have a Visa or Master Card credit card.

1. How many students have aMaster Card?

a. 30 b. 40 c. 55 d. 452. How many students have

neither a Visa nor a MasterCard?

a. 30 b. 40 c. 55 d. 45

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 4 / 27

Popper 01 Questions

Find the indicated set using the folloiwng information:

U = {1,2,3,4,5,6,7,8,9,10}

A = {1,2,5,6,9,10}, B = {3,4,7,8}, C = {2,3,8,9,10}

3. Ac

a) {1,2,5,6,9,10}b) {2,3,4,7,8,9,10}c) {3,4,7,8}d) {∅}

4. A ∩ (C ∩ Bc)c

a) {1,5,6}b) {2,3,8,9,10}c) {1,2,5,6,9,10}d) {2,9,10}

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 5 / 27

Popper 01 Questions

5. You have 5 tickets to a Texan’s game. In how many ways can youtake your best friend and 3 other friends from a group of 6 friends?

a) 15

b) 18

c) 20

d) 22

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 6 / 27

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 ways E can occurs

total number of possible observations

=n(E)

n(S)

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 7 / 27

Example 1

Suppose a box contains 3 defective light bulbs and 12 good bulbs.Suppose we draw a simple random sample of 4 lightbulbs, find theprobability that one of the bulbs drawn is defective.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 8 / 27

Example 1 continued

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

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

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

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 9 / 27

Example 2

Suppose we select randomly 4 marbles drawn from a bag containing 8white and 6 black marbles.

1. What is the probability that half of the marbles drawn are white?

2. What is the probability that at least 2 of the marbles drawn arewhite?

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 10 / 27

Example 3

A sports survey taken at UH shows that 48% of the respondents likedsoccer, 66% liked basketball and 38% liked hockey. Also, 30% likedsoccer and basketball, 22% liked basketball and hockey, and 28% likedsoccer and hockey. Finally, 12% liked all three sports.

1. What is the probability that a randomly selected student likesbasketball or hockey?

2. What is the probability that a randomly selected student does notlike any of these sports?

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 11 / 27

Probability of an Event

The probability of an event E is calculated by summing theprobabilities of the sample points in the sample space for E .Notation: P(E)

In the insurance example, the probability that a driver has anynumber of accidents in a given year is: P(0) = 0.25, P(1) = 0.45,P(2) = 0.20, P(3 or more) = 0.10.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 12 / 27

Basic Rules of Probability

1. For any event E , 0 ≤ P(E) ≤ 1.

2. For the universal set, U, P(U) = 1.

3. Complement Rule: For any event E , P(Ec) = 1− P(E) and forany two events, E and F , P(E ∩ F c) = P(E)− P(E ∩ F ).

4. General Addition Rule: For any two events E and F ,P(E ∪ F ) = P(E) + P(F )− P(E ∩ F ).

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 13 / 27

Rules of Probability, 5

General rule for multiplication: For any two events E and F

P(E ∩ F ) = P(E)× P(F |E)

orP(E ∩ F ) = P(F )× P(E |F )

Where P(F |E) is the probability of F given that the event E hasoccurred. Similarly P(E |F ) is the probability of E given that F hasoccurred. These types of probabilities are called conditionalprobability.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 14 / 27

Example 3

Choose a person at random. Let A be the event that a person chosenis a women, and B the event that the person holds a managerial orprofessional job. From Government data P(A) = 0.46 and theprobability of managerial and professional jobs among women isP(B|A) = 0.32. What is the probability that a randomly chosen personis a woman and holds a managerial or professional job?

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 15 / 27

Example 4

Thirty percent of the students at a local high school face a disciplinaryaction of some kind before they graduate. Of those "felony" students,40% go on to college. Of the ones who do not face disciplinary action60% go on to college.

1. What is the probability that a randomly selected student bothfaced a disciplinary action and went on to college? P(F ∩ C).

2. What percent of the students from the high school go on tocollege?

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 16 / 27

Example 5

A person must select one of three boxes, each filled with toy cars. Theprobability of box A being selected is 0.19, of box B being selected is0.18, and of box C being selected is 0.63. The probability of finding ared car in box A is 0.2, in box B is 0.4, and in box C is 0.9. We areselecting one of the toy cars.

1. What is the probability that the toy car is red and in box A?

2. What is the probability that the toy car is red and in box B?

3. What is the probability that the toy car is red and in box C?

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 17 / 27

Example 6

At a Ford dealership, if you select a Ford Mustang at random, theprobability it is red is P(R) = 0.40, the probability it is convertible P(C) =0.14, and the probability that it is red or a convertible P(R or C) = 0.50.

1. What is the probability that a randomly selected Ford Mustang is ared convertible?

2. What is the probability that a randomly selected Ford Mustang isnot red but a convertible?

3. What is the probability of getting a convertible out of the redMustangs?

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 18 / 27

Using the Venn Diagram

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 19 / 27

Conditional Probability

For any two events, A and B, the probability of getting event A, given(given B has occurred, out of B, if B has occurred ) B is the probabilityof A and B divided by the probability of B.

P(A|B) =P(A ∩ B)

P(B)

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 20 / 27

Example 7

Suppose we draw two cards from a deck of 52 fair playing cards, whatis the probability of getting an ace on the first draw and a king on thesecond draw?

Without replacement.

With replacement.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 21 / 27

Independent Events

Two events E and F are independent if the outcome of one eventdoes not changes the probability of the other event.If E and F are independent then,

P(E |F ) = P(E)andP(F |E) = P(F )

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 22 / 27

Example 8

Determine if A and B are independent.1. P(A) = 0.9, P(B) = 0.3, P(A ∩ B) = 0.27

2. P(A) = 0.4, P(B) = 0.6, P(A ∩ B) = 0.20

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 23 / 27

Example 9

A clothing store targets young customers (ages 18 through 22) wishesto determine whether the size of the purchases related to the methodpayment. Suppose a customer is picked at random. The following is300 customers the amount of the purchase and method payment.

Cash Credit Layaway TotalUnder $40 60 30 10 100

$40 or more 40 100 60 200Total 100 130 70 300

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 24 / 27

a) What is the probability that the customer paid with a credit card?

b) What is the probability that the customer purchased under $40?

c) What is the probability that the customer paid with credit card giventhat the purchase was under $40?

d) What is the probability that the customer paid with credit card andthat the purchase was under $40?

e) Are type of payment and amount of purchase independent?

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 25 / 27

Two Frequently Asked Questions

1. When do I add and when do I multiply?I Add when finding the chance of events A or B or both happening.

P(A or B) = P(A ∪ B) = P(A) + P(B)− P(A ∩ B)

I Multiply when finding the chance that both events A and B happen.

P(A and B) = P(A ∩ B) = P(A)× P(B, given A) = P(A)P(B|A)

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 26 / 27

Two Frequently Asked Questions

2. What’s the difference between disjoint and independent?I Two events are disjoint if the occurrence of one prevents the other

from happening.P(A ∩ B) = 0

I Two events are independent if the occurrence of one does notchange the probability of the other.

P(A|B) = P(A)

Cathy Poliak, Ph.D. cathy@math.uh.edu Office: Fleming 11c (Department of Mathematics University of Houston )Section 2.3 & 2.4 lecture 5 - 2311 27 / 27