QBA 282 Ch. 4-5 Exam Study Set

discrete uniform probability distribution

a probability distribution for which each possible value of the random variable has the same probability

random variable

a numerical description of the outcome of an experiment

Addition Law

a probability law used to compute the probability of the union of two events

Even though the experiment is repeated in exactly the same way,

an entirely different outcome may occur.

multi-step experiment

an experiment that can be described as a sequence of steps

experiment

any process that generates well-defined outcomes.

statistical experiment

any random activity that results in a definite outcome.

Relative Frequency Method

assigning probabilities based on experimentation or historical data

Twenty percent of people at a company picnic got food poisoning. What percent of the people at the picnic did NOT get food poisoning? Select one: a. 20% b. 80% c. 40% d. 60%

b. 80%

The multiplication law is potentially helpful when we are interested in computing the probability of _____. Select one: a. mutually exclusive events b. the intersection of two events c. the union of two events d. None of the above

b. the intersection of two events

The symbol ∪ indicates the _____. a. intersection of events b. union of events c. sample space d. sum of the probabilities of events

b. union of events

Suppose we flip a fair coin five times and each time it lands heads up. The probability of landing heads up on the next flip is _____. Select one: a. 1 b. 0 c. 1/2

c. 1/2

Classical Method of Assigning Probabilities

used when an experiment has equally likely outcomes

An experiment has three steps, with three outcomes possible for the first step, two outcomes possible for the second step and four for the third step. How many outcomes exist for this experiment?

(3)(2)(4) = 24 outcomes (use multi-step experiment)

If an experiment consists of a sequence of k steps in which there are n1 possible results for the first step, n2 possible results for the second step, and so on, then the total number of experimental outcomes is given by what?

(n1)(n2) . . . (nk)

Given the following distribution, find E(x): x f(x) 3 .25 6 .50 9 .25

3/0.25 = 0.75; 6/0.50 = 3; 9/0.25= 2.25 0.75+3+2.25 = 6

event

A collection of sample points

probability function

A function, denoted f(x), that provides the probability that a discrete random variable x takes on some specific value.

Probability

A numerical measure of the likelihood that an event will occur.

Binomial Probability Distribution

A probability distribution showing the probability of x successes in n trials of a binomial experiment.

Union of Events

All events that are in A or B or both (everything!); A∪B; ∪= or/union

Counting Rule for Permutations

Allows one to compute the number of experimental outcomes when n objects are to be selected from a set of N objects WHERE THE ORDER OF SELECTION IS IMPORTANT The same n objects selected in a different order are considered a different outcome.

sample point

An experimental outcome and an element of the sample space.

The Powerball lottery is played twice each week in 31 states, the District of Columbia, and the Virgin Islands. To play Powerball, a participant must purchase a $2 ticket, select five numbers from the digits 1 through 59, and then select a Powerball number from the digits 1 through 35. To determine the winning numbers for each game, lottery officials draw five white balls out a drum of 59 white balls numbered 1 through 59 and one red ball out of a drum of 35 red balls numbered 1 through 35. To win the Powerball jackpot, a participant's numbers must match the numbers on the five white balls in any order and must also match the number on the red Powerball. The numbers 5-16-22-23-29 with a Powerball number of 6 provided the record jackpot of $580 million ( Powerball website, November 29, 2012). How many Powerball lottery outcomes are possible? (Hint: Consider this a two-step experiment. Select the five white ball numbers and then select the one red Powerball number.)

C 59,5/C 35,1 = (59)(58)(57)(56)(55)/(5)(4)(3)(2)(1) = 5,006,386 * 35 = 175,223,510 outcomes

KP&L has designed a project to increase the generating capacity of one of its plants. The project is divided into two sequential stages : Stage 1 - Design; Stage 2 - Construction. The possible completion times of these stages are as follows: Possible completion time in months Design Stage Construction Stage 2 6 3 7 4 8

Design stage n1 = 3 Construction stage n2 = 3 Total Number of Experimental Outcomes: n1n2 = (3)(3) = 9

Expected value for binomial probability distribution

E(x) = np

Promotion status of police officers over the past two years: Men Women Total Promoted 288 36 324 Not Promoted 672 204 876 Total 960 240 1200 Men (M) Women (W) Total Probability Table Promoted (A) 288/1200 = 0.24 0.03 0.27 Not Promoted (Ac ) 0.56 0.17 0.73 Total 0.80 0.20 1.00

Event A = An officer is promoted Event M = The promoted officer is a man P(A │ M) = An officer is promoted given that the officer is a man P(A│M)= (P(A∩M))/(P(M)) From the table we know: P(A∩M) = 0.24 P(M) = 0.8 P(A│M)=0.24/0.8 = 0.3

If an experiment has n possible outcomes, the classical method would assign a probability of 1/n to each outcome. What would be the experiment, sample space, and probabilities?

Experiment: Rolling a die Sample Space: S = {1, 2, 3, 4, 5, 6} Probabilities: Each sample point has a 1/6 chance of occurring

two types of discrete probability distribution

First type: uses the rules of assigning probabilities to experimental outcomes to determine probabilities for each value of the random variable. Second type: uses a special mathematical formula to compute the probabilities for each value of the random variable.

Relative Frequency method formula

Frequency/total frequency

Counting Rule for Multiple Step Experiments

If an experiment consists of a sequence of k steps in which there are n1 possible results for the first step, n2 possible results for the second step, and so on.

A small assembly plant with 50 employees is carrying out performance evaluation. Each worker is expected to complete work assignments on time and in such a way that the assembled product will pass a final inspection. The results were as follows: Result Number of Employees Relative Frequency Late completion of work 5 5/50 = 0.1 Assembled a defective work 6 0.12 Completed work late and assembled defective work 2 0.04 Event L = the event that the work is completed late Event D = the event that the assembled product is defective The production manager decided to assign poor performance rating to any employee whose work is either late or defective.

L ∪ D = Event that the production manager assigned an employee a poor performance rating P(L ∪ D) = P(L) + P(D) - P(L ∩ D) P(L ∪ D) = 0.10 + 0.12 - 0.04 = 0.18

Use of credit card for purchase of gasoline From past experience it is known 80% of customers use credit card for the purchase of gasoline. The service station manger wants to determine the probability that the next two customers purchasing gasoline will each use a credit card. Event A: First customer uses a credit card Event B: The second customer uses a credit card

P(A∩B) = P(A)P(B) = (0.8) (0.8) = 0.64

An experiment with three outcomes has been repeated 50 times. It was learned that E1 occurred 20 times, E2 occurred 13 times and E3 occurred 17 times. Assign probabilities to the outcomes given.

P(E1) = 2/5 = 0.4 = 40% P(E2) = 13/50 = 0.26 = 26% P(E3) = 17/50 = 0.34 = 34%

Newspaper circulation department Event D = A household subscribes to the daily edition Event S = The household already holds subscription to the Sunday edition Given: P(D) = 0.84 P(S|D) = 0.75 What is the probability that the household subscribes to both the Sunday and daily editions of the newspaper?

P(S∩D) = P(D)P(S|D) = (0.84) (0.75) = 0.63

Statistical experiments are sometimes called what?

Random experiments

complement of event

The complement of event A consists of all outcomes that are NOT in A.

If the probability of the likelihood of occurrence is at 1, what does that mean?

The event is almost certain to occur.

If the probability of the likelihood of occurrence is at 0, what does that mean?

The event is very unlikely to occur.

If the probability of the likelihood of occurrence is at 0.5, what does that mean?

The occurrence of the event is just as likely as it is unlikely.

assigning probabilities

The probability assigned to each experimental outcome must be between 0 and 1, inclusively. 0 < P(Ei) < 1 for all i • where: Ei is the ith experimental outcome P(Ei) is its probability 2. The sum of the probabilities for all experimental outcomes must equal 1. P(E1) + P(E2) + . . . + P(En) = 1 where: n is the number of experimental outcomes

Intersection of Events

The probability that Events A and B both occur is the probability of the intersection of A and B. The probability of the intersection of Events A and B is denoted by P(A ∩ B). If Events A and B are mutually exclusive, P(A ∩ B) = 0. The probability that Events A or B occur is the probability of the union of A and B.

subjective method

a method of assigning probabilities on the basis of judgment

expected value

The weighted average of a random variable.

mutually exclusive events

Two events that cannot occur at the same time

independent events

Two or more events in which the outcome of one event does not affect the outcome of the other event(s).

probability distribution

a description of how the probabilities are distributed over the values of the random variable

tree diagram (probability)

a diagram used to show the total number of possible outcomes in an experiment

Venn Diagram

a graphical representation for showing symbolically the sample space and operations involving events in which the sample space is represented by a rectangle and events are represented as circles within the sample space.

continuous random variable

a random variable that may assume any numerical value in an interval or collection of intervals

discrete random variable

a random variable that may assume either a finite number of values or an infinite sequence of values

The following probability distributions of job satisfaction scores for a sample of IS senior executives and managers range to a low of 1 to a high of 5. Probability Job Satisfaction Score IS Senior Executives IS Managers 1 .05 .04 2 .09 .10 3 .03 .12 4 .42 .46 5 .41 .28 a. What is the expected value of job satisfaction for Senior Executives? b. What is the expected value of job satisfaction for Managers? c. What is the probability that a senior executive has a job satisfaction score of less than 3? d. What is the probability that a senior executive has a job satisfaction score of 3 or less?

a. (1)(0.05)+(2)(0.09)+(3)(0.03)+(4)(0.42)+(5)(0.41) = 4.05 b. (1)(0.04)+(2)(0.10)+(3)(0.12)+(4)(0.46)+(5)(0.28) = 3.84 c. P(less than 3) = 0.05+0.09 = 0.14 d. P(3 or less) = 0.05+0.09+0.03 = 0.17

Probability values are always assigned on a scale from: Select one: a. 0-1, inclusive b. 0-10, inclusive c. 0-100, inclusive d. 1-100, inclusive

a. 0-1, inclusive

High school seniors with strong academic records apply to the nation's most selective colleges in greater numbers each year. Because the number of slots remains relatively stable, some colleges reject more early applicants. Suppose that for a recent admissions class, an Ivy League college received 2851 applications for early admission. Of this group, it admitted 1033 students early, rejected 854 outright, and deferred 964 to the regular admission pool for further consideration. In the past, this school has admitted 18% of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was 2375. Let E, R, and D represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool. a. Use the data to estimate P(E), P(R), and P(D). b. Are events E and D mutually exclusive? Find P(E∩D).

a. 1033/2851 = 0.3623; 854/2851 = 0.2995; 964/2851 = 0.3381 b. Yes, because students cannot be admitted for the early admission and deferred at the same time. 0/2851 = 0

Junior Achievement USA and the Allstate Foundation surveyed teenagers aged 14 to 18 and asked at what age they think they will become financially independent. The responses of 944 teenagers who answered this survey question are as follows. Age Financially Independent Number of Responses 16 to 20 191 21 to 24 467 25 to 27 244 28 or older 42 a. Compute the probability of being financially independent for each of the four age categories. b. What is the probability of being financially independent before age 25?

a. 191/944 = 0.2023; 467/944 = 0.4947; 244/944 = 0.2585; 42/944 = 0.0445 b. 658/944 = 0.6970

Employee retention is a major concern for many companies. A survey of Americans asked how long they've worked for their current employer (BLS website 2015). Consider the following sample data of 2000 employees who graduated from college five years ago. Time in years with Current employer Number 1 506 2 390 3 310 4 218 5 576 Let x be the random variable indicating the number of years the respondent has worked for his/her employer. a. Use the data to develop a discrete probability distribution for x. b. What is the probability that a respondent has been at his/her place of employment for more than three years?

a. 506/2000 = 0.253; 390/2000 = 0.195; 310/2000 = 0.155; 218/2000 = 0.109; 576/2000 = 0.288 0.253+0.195+0.155+0.109+0.288 = 1 b. P for more than 3 years = P(x+4)+P(x+5) = 0.109+0.288 = 0.397 or 39.7%

Fortune magazine publishes an annual list of the 500 largest companies in the United States. The corporate headquarters for the 500 companies are located in 38 different states The following table shows the eight states with the largest number of Fortune 500 companies (Money/CNN website, May 12, 2012). State Number of Companies California 53 Illinois 32 New Jersey 21 New York 50 Ohio 28 Pennsylvania 23 Texas 52 Virginia 24 Suppose one of the 500 companies is selected at random for a follow-up questionnaire. a. What is the probability that the company selected has its corporate headquarters in California? b. What is the probability that the company selected has its corporate headquarters in California, New York, or Texas?

a. 53/500 = 0.106 b. 155/500 = 0.31

Kentucky Power and Light Company (KP&L) We have completion results for 40 KP&L projects. Completion time in months Design Construction Sample Point Number of past projects with this completion time Probability 2 6 (2,6) 6 0.15 2 7 (2,7) 6 0.15 2 8 (2,8) 2 0.05 3 6 (3,6) 4 0.10 3 7 (3,7) 8 0.20 3 8 (3,8) 2 0.05 4 6 (4,6) 2 0.05 4 7 (4,7) 4 0.10 4 8 (4,8) 6 0.15 a. Event C = Project will take 10 months or less to complete b. Event L = Project will take less than 10 months to complete

a. C = { (2,6), (2,7), (2,8), (3,6) , (3,7), (4,6)} P (C )= { P(2,6) + P (2,7) + P (2,8) + P (3,6) +P (3,7) +P (4,6)} P(C) = 0.15 + 0.15 + 0.05 + 0.10 + 0.20 + 0.05 = 0.7 b. C = { (2,6), (2,7), (3,6)} P (C )= { P(2,6) + P (2,7) + P (3,6)} P(C) = 0.15 + 0.15 + 0.10 = 0.4

Consider the experiment of tossing a coin twice. a. List the experimental outcomes. b. Define a random variable that represents the number of tails occurring on the two tosses. c. Show what value the random variable would assume for each of the experimental outcomes. d. Is this random variable discrete or continuous?

a. HH, HT, TH, TT b. x = number of heads occurring on two tosses with a coin. c. Outcome HH HT TT X 2 1 1 0 d. discrete

Suppose that we have two events, A and B, with P(A)= .50,P(B)= .60 and P(A∩B)= .40. a. Find P(A | B) b. Find P(B | A)

a. P(A | B) = P(A and B)/P(B) = 0.40/0.60 =2/3 = 0.6667 = 66.67% b. P(B | A) = P(A and B)/P(B) = 0.40/0.50 = 4/5 = 0.8 = 80%

The following table shows information collected from 10 airlines regarding arrivals, baggage and complaints. Airline On-time arrivals (%) Mishandled baggage per 1000 passengers Customer Complaints per 1000 passengers Virgin America 83.5 .87 1.5 JetBlue 79.1 1.88 .79 AirTran 87.1 1.58 .91 Delta 86.5 2.10 .73 Alaska Airlines 87.5 2.93 .51 Frontier 77.9 2.22 1.05 Southwest 83.1 3.08 .25 US Airways 85.9 2.14 1.74 American 76.9 2.92 1.8 United 77.4 3.87 4.24 a. If you randomly choose a Delta Air Lines flight, what is the probability that this individual flight has an on-time arrival? b. If you randomly choose one of the 10 airlines for a follow-up study on airline quality ratings, what is the probability that you will choose an airline with less than two mishandled baggage reports per 1000 passengers? c. If you randomly choose 1 of the 10 airlines for a follow-up study on airline quality ratings, what is the probability that you will choose an airline with more than one customer complaint per 1000 passengers? d. What is the probability that a randomly selected AirTran Airways flight will not arrive on time?

a. P(on-time arrival) = 86.5% = 0.865 b. P(less than 2) = 3/10 = 0.30 c. P(more than 1) = 5/10 = 0.50 d. P(not on time) = 1-P(on time) = 1- .871 = 0.129

A joint survey by Parade magazine and Yahoo! found that 59% of American workers say that if they could do it all over again, they would choose a different career (USA Today, September 24, 2012). The survey also found that 33% of American workers say they plan to retire early and 67% say they plan to wait and retire at age 65 or older. Assume that the following joint probability table applies. Retire Early Yes No Career Same .20 .21 .41 Different .13 .46 .59 .33 .67 a. What is the probability a worker would select the same career? b. What is the probability a worker who would select the same career plans to retire early?

a. P(same career) = 0.41 b. P(retire early | same career) = P(retire early and same career)/P(same career) = 0.20/0.41 = 0.4878

Clarkson University surveyed alumni to learn more about what they think of Clarkson. One part of the survey asked respondents to indicate whether their overall experience at Clarkson fell short of expectations, met expectations, or surpassed expectations. The results showed that 4% of the respondents did not provide a response, 26% said that their experience fell short of expectations, and 65% of the respondents said that their experience met expectations. a. If we chose an alumnus at random, what is the probability that the alumnus would say their experience surpassed expectations? b. If we chose an alumnus at random, what is the probability that the alumnus would say their experience met or surpassed expectations?

a. P(surpassed) = 100% - 4% - 26% - 65% = 5% = 0.05 b. P(met or surpassed) = P(met) + P(surpassed) = 65% + 5% = 70% = 0.70

For each of the following variables, label as discrete or continuous. If discrete, label as finite or infinite: a. Number of questions answered correctly on your individual Moodle quiz b. Number of cars driving through the drive through at Raising Cane's in an hour c. Weight in pounds from a shipment of pencils Time between arrivals (in minutes) for a toll booth

a. discrete; finite b. discrete; infinite c. continuous d. continuous

The probability of an event is ____________. a. the sum of the probabilities of the sample points in the event b. the product of the probabilities of the sample points in the event c. the minimum of the probabilities of the sample points in the event d. the maximum of the probabilities of the sample points in the event

a. the sum of the probabilities of the sample points in the event

Counting Rule for Combinations

enables us to count the number of experimental outcomes when n objects are to be selected from a set of N objects

Calculating Expected Value

f * f(x) = xf(x)

required conditions for a discrete probability function

f(x) > 0 and f(x) = 1

Calculation for discrete probability distribution

n/total

Multiplication Law

provides a way to compute the probability of the intersection of two events

Conditional Probability

the probability that one event happens given that another event is already known to have happened

random experiment

the process of obtaining information from data collected from a study that has unpredictable outcomes

sample space

the set of all possible outcomes of an experiment