Data Analytics: Chapter 5: Probability

conditional probabilities

found by restricting ourselves to a single row or column

special law of addition

if A and B are mutually exclusive events, then P(A n B) = 0 and the general addition law can be simplified to the sum of the individual probabilities for A and B - addition law for mutually exclusive events

Define event A = (1, 2, 3, 4) and event B =(2, 3, 6, 7). A and B =

(1, 2, 3, 4, 6, 7)

The odds against a horse winning a race were set at 7 to 1. The probability of that horse NOT winning the race is _____. (answer should be in decimal form, using 3 decimal places)

.0875

80% of companies ship their products by truck. 50% of companies ship their product by rail. 60% of companies ship by either truck or rail. What is the probability that a company ships by truck and rail?

.7

P(A) = .45 and P(B)=.2. A and B are mutually exclusive. P(a given B) =

0 because they are mutually exclusive

In order to make sure Sam gets to his final exam on time he sets 3 alarms that work independently of each other. Assume the probability of any one of the alarms working is equal to .98. What is the probability that Sam is late to his final exam?

0.000008

A manufacturer of liquid laundry detergent has a .02 probability that the detergent bottles will be improperly filled. (Either too much or too little detergent). There is a .03 probability that the label on the bottle will not be affixed properly. If the events of bottle fill and affixing the label are independent, then the probability of a bottle being filled improperly and having an improperly affixed label is _______. (round your answer to four decimal places).

0.02 * 0.03=0.0006

The probability of an employee getting a promotion is 0.2. The probability of an employee having an MBA is 0.3. The probability of an employee getting a promotion given that the employee has an MBA is 0.28. What is the probability that an employee has an MBA and gets a promotion.

0.28*0.3 = 0.084

The probability of Margaret getting a promotion a XYZ Corp is 0.7. The probability of Katia getting a promotion at ABC Inc is 0.6. If the two promotions are unrelated, then the probability of both Margaret and Katia receiving a promotion is ________. (round to 2 decimal places).

0.7*0.6=0.42

The sum of the probabilities of all the outcomes in the sample space is __________.

1

XYZ Corp. has filled 100,000 purchase orders during its existence. 1,100 of the purchase orders have had errors. Using and empirical probability, the probability of the next purchase order having an error is _________.

1,100/100,000=.011

The probability of a customer purchasing popcorn at the movie theater is 0.3. What is the probability that a customer will not purchase popcorn?

1-0.3 = 0.7

Which of the following events are mutually exclusive?

1. Being on time and being late for an appointment 2. rolling an odd number and an even number on the same roll of a die

The multiplication rule is used to calculate what type of probability?

1. conditional 2. joint

Assume P(A) = .3 and P(B) = .4. If the P(A or B) =.7 we can say that A and B are

1. disjoint 2. mutually exclusive

The people who are involved in which of the following areas talk more commonly about odds rather than speaking of probability:

1. games of chance 2. sports

The sports book at the High Roller Casino put the odds of a certain baseball team to win the World Series at 1:25 (1 to 25). Based on those odds, what is the probability that this baseball team will win the World Series?

1/(1+25) = .0385

The odds against a horse winning a race were set at 7 to 1. The probability of that horse winning the race is ______. (answer should be in decimal form, using 3 decimal places.)

7+1=8 1/8 = .125

Kareem is trying to decide which college to attend full time next year. Kareem believes there is a 55% chance that he will attend State College and a 33% chance that he will attend Northern University. The probability that Kareem will attend either State or Northern is _______. (state your answer as a decimal and round your answer to two decimal places.)

P (A or B) = P(A) + P(B) P (A or B) = .55 + .33 = .88

If the events A and B were independent and P(A) = .4 and P(A and B) =.2 then P(B) = _____

P(A and B)= P(A)*P(B) .2 = .4*P(B) divide .4 from both sides .5 = P(B)

Given P(A) = .4, P(B) = .6, and P(A and B)=.2, calculate the conditional probability of A given B.

P(A given B) = P(A and B)/ P(B) P(A given B) = .2/.6 P(A given B) = .33333....

The probability of a customer ordering popcorn at the movie theater is .4. The probability of a customer ordering a drink at the movie theater is .65. The probability of a customer ordering popcorn and a drink is .3. If a customer has already ordered a drink, what is the probability the customer will order popcorn?

P(A given B)= P(A and B)/ P(B) P(A given B) = .3/.65 P(A given B) = 0.46

The probability of Anthony being on time for work is 0.9. The probability that Anthony will take the train to work is 0.8. The probability that Anthony will be on time fro work if he took the train is 0.95. The probability that Anthony is on time fro work and took the train is ________. ( 2 decimal places).

P(Train and On time) = P(Train) x P(On Time given Train) P(Train and On time) = .8*.95=.76

combination

a collection of r items chosen at random without replacement from n items where th order of the selected items is not important excel function: =COMBIN(n,r)

contingency table

a cross-tabulation of frequencies into rows and columns - often used to report the results of a survey

probability

a number that measures the relative likelihood that the event will occur. The probability of an event A, denoted P(A), must like within the interval from 0 to 1. 0 = the event is not likely to occur 1 = the event is certain to occur in a discrete sample space, the probabilities of all simple events must sum to 1 because it is certain that one of them will occur.

marginal probability

a relative frequency that is found by dividing a row or column total by the total sample size

simple event / elementary event

a single outcome

permutation

an arrangement of the r sample items in a particular order excel function: =PERMUT(n,r)

random experiment

an observational process whose results cannot be known in advance

event

any subset of outcomes in the sample space

law of large numbers

as the number of trials increases, any empirical probability approaches its theoretical limit

A probability that can be deduced through logical reasoning before an experiment is performed is what type of probability?

classical

Events are considered ________ ________ if the union of these events is the entire sample space.

collectively exhaustive

Events that include more than one outcome from the sample space are known as ________ events.

compound

compound event

consisting of two or more simple events

union of two events

consists of all outcomes in the sample space S that are contained either in event A or in event B or in both

complement of an event

denoted as A' and consists of everything in the sample space S except event A - Since A and A' together comprise the sample space, their probabilities sum to 1 - the probability of the complement of A is found by subtracting the probability of A from 1

The probability of Murali going to the coffee shop is 0.7. The probability of Connie going to the coffee shop is 0.4. If Muarli goes to the coffee shop, the probability of Connie going to the coffee shop is 0.48. The events Murali going to the coffee shop and Connie going to the coffee shop are _______.

dependent

A coach observes Tom making 10 free throws out of his last 30 attempts. Using the _________ approach the coach assigns a probability of _______ to Tom making his next free throw. (Round probability to one decimal place).

empirical; 10/30 = .3

Using the multiplication rule, the joint probability of event A and event B is computed by multiplying the conditional probability of event A given event B by the probability of ________.

event B

mutually exclusive / disjoint

events A and B are mutually exclusive if their intersection is the empty set (a set that contains no elements)

collectively exhaustive events

events are collectively exhaustive is their union is the entire sample space S - two mutually exclusive, collectively exhaustive events are binary (or dichotomous events)

True or false: Define event A = (1, 2, 3, 4) and event B = (2, 3, 6, 7). A u B = (2, 3).

false

True or false: The General Law of Multiplication is used to calculate the probability of the union of two events.

false -- it gets the intersection

Which of the following is not an example of an experiment?

pick the team that won last years World Series

a priori

refers to the process of assigning probabilities before we actually observe the event or try and experiment

subjective approach to probability

reflect's someones informed judgement about the likelihood of an event - needed when there is no repeatable random experiment

joint probabilities

represent the intersection of two events

A contingency table is sometimes called an r x c tables where r stands for number of ______ and c stands for number of ______.

rows; columns

The set of all possible outcomes from a random experiment is called a _____ ______.

sample space

general law of addition

says the probability of the union of two events A and B is the sum of their probabilities less the probability of their intersection

A softball coach believes that Laurie has a .3 probability of getting hit against a left-handed pitcher that Laurie has never battled against before. The .3 probability that the coach has assigned is what type of probability?

subjective

intersection of two events

the event consisting of all outcomes in the sample space S that are contained in both event A and event B

sample space

the set of all possible outcomes (denoted as S)

empirical or relative frequency approach to probability

use when collecting empirical data through observations or experiments - assign probabilities by counting the frequency of observed outcomes (f) defined in our experimental sample space and dividing by the number of observations (n)

classical approach to probability

using deduction to determine P(A) - based on logic or theory, not an experience. - rarely possible in business situations

If two events are mutually exclusive then the probability of their intersection is _____.

zero

Probability values range from _____ to ______.

zero; one

The probability of State college winning a football game is .6. The probability of University of State winning a football game is .65. Given that state college has won its football game, the probability of university of state winning its game is .65. The teams are not playing each other. The events State College winning and University of State winning are _____

independent

The ______ (one word) of two events A and B contains only those outcomes that are in both A and B.

intersection

Suppose in a population of adults there are 10% that are avid fly fishermen/women. If we were to choose a random sample of 10 adults from this population the law of large numbers says ______.

that the percent of fishermen/women in this sample might not equal 10% but as the sample gets larger the percent will get closer to 10%

factorial

the number of unique ways that n items can be arranged in a particular order - the product of all integers from 1 to n excel function: =FACT(n!)

conditional probability

the probability of event A giving that event B has occurred - denoted as P(A ∣ B), which is read "the probability of A given B

To calculate a joint probability from a contingency table, the frequency of each cell is divided by the _______.

total number of outcomes in the sample space