Deck 6: Probability

Question 1

Two events are said to be disjoint or mutually exclusive when they have no outcomes in common.

Answer 1

True

Question 2

Any collection of possible outcomes of a chance experiment is called a sample space.

Answer 2

False

Question 3

A chance experiment is any activity or situation in which there is uncertainty concerning which of two or more possible outcomes will result.

Answer 3

True

Question 4

The probability of an event E can always be computed using the formula, ​

Question 5

With the advent of cable television, a tremendous number of channels and wide variety of programs are available for viewing. Consider an experiment where the weekly TV listings are located, a random page is chosen, and a random point on the page is picked. The outcome of the experiment is whatever program is described at that point on the page. Define events P₁, P₂, and P₃ as follows: P₁ = the program is an educational program P₂ = the program is on a regular network channel, not cable P₃ = the program is intended for children For each problem below, sketch a standard Venn diagram with 3 circles and shade the regions that correspond to the following events:
a) An educational program for adults is selected.
b) A children's program is selected.
c) A network educational program for children is selected.

Question 6

Consider the chance experiment in which a high school principal selects a sports event at random to attend. The sex of the participants (Male or Female) is noted, as well as the type of event (Wrestling, Volleyball, Basketball, or Swimming).
a) Display the possible outcomes of this selection using a tree diagram.
b) Let M be the event that the participants are male and B be the event that a ball is involved (i.e. basketball or volleyball).
i) What outcomes are in B^C?
ii) What outcomes are in ?
iii) What outcomes are in ?

Question 7

A statistics class conducts a chance experiment in which they observe students who buy a snack and the school vending machines. The type of food (Liquid, or Solid) is noted, as well as the number of calories of fat content (25, 50, or 75).
a) Display the possible outcomes using a tree diagram.
b) Let L be the event that the item is liquid and B be the event that the item has fewer than 75 calories of fat.
i) What outcomes are in B^C?
ii) What outcomes are in ?
iii) What outcomes are in ?

Question 8

If two events, A and B, are disjoint, then .

Question 9

The classical view of probability is based on the Law of Large Numbers.

Question 10

In order to ensure the safety of school classrooms the local Fire Marshall does an inspection at Thomas Jefferson High School every month, looking for faulty wiring, overloaded circuits, etc. At TJHS the new Academic Wing has 5 math rooms, 10 science rooms, and 10 English rooms. The science rooms are divided into 8 biology and 2 chemistry rooms. Each month, the Fire Marshall randomly picks one of the rooms in the new wing to inspect each month. Define the following events: S = the event the selected room is a science room B = the event the selected room is a biology room M = the event the selected room is a math room E = the event the selected room is an English room C = the event the selected room is a chemistry room Calculate the probabilities of the events described below:
a) P(S)
b) P(M or E )
c) P(E or B) d) P(S and not C)

Question 11

Two events are independent if they cannot occur simultaneously.

Question 12

As it gets close to time for school to start, automobiles begin to arrive on campus. Consider an experiment where a vehicle is selected at random. Define events as follows: V₁ = the vehicle belongs to a school staff member V₂ = the vehicle is less than 5 years old V₃ = the vehicle is American made For each problem below, sketch a standard Venn diagram with 3 circles and shade the regions that correspond to the following events: a) An American vehicle less than 5 years old is selected.
b) A staff member's vehicle is selected.
c) An American made staff member's vehicle at least 5 years old is selected.

Question 13

A small manufacturing firm has 250 employees. Fifty have been employed for less than 5 years and 125 have been with the company for over 10 years. Suppose that one employee is selected at random from a list of the employees. For the following events, compute the probabilities requested below. A = the event the selected employee has been with the firm less than 5 years B = the event the selected employee has been with the firm 5 ≤ x ≤ 10 years. C = the event the selected employee has been with the firm over 10 years. ​
a) P(A)
b) P(C)
c) P(A or B) d) P(A and C)

Question 14

The General Addition Rule states that ​

Question 15

Two events, E and F, are independent if .

Question 16

In a few sentences, distinguish between the classical and relative frequency approaches to probability.

Question 17

An event consisting of exactly one outcome is called a simple event.

Question 18

The event "A or B" consists of all of the outcomes in both of the events.

Question 19

In a few sentences, define the following terms: a) Sample space b) Disjoint events

Question 20

The General Multiplication Rule states that .

Question 21

The football team at North Snowshoe High is in a little bit of trouble. Two of their players have just left the game with injuries, and due to the flu the coach only has 5 players to choose from: three linemen, a quarterback, and the punter who doubles as the team statistician. The coach must choose two of those five to go into the game. Unfortunately, it is snowing very severely, making it difficult for the coach to see the players on the bench. Therefore he randomly selects two players to go into the game. (He CAN tell there are two players, so of course he sends them in without replacement!) What is the probability that the team statistician will be selected?

Question 22

Exhibit 6-1 In a survey of airline travelers, subjects were observed in the coach section of airplanes to determine if men or women are bothered by a seatmate of the opposite gender using the common armrest. ("Passengers who were asleep or lovers cuddled together were not counted, since these were considered confounding circumstances.") The table below contains data gathered in that study. Suppose one of these passengers was randomly selected. Use the information in the table to answer the questions below. In showing your work, define and use appropriate notation.
In the study in Exhibit 6-1, passengers were also classified by age: Suppose one of these passengers was randomly selected. Calculate the probability that:
a) The passenger is under 40, given that she is female.
b) The passenger was bothered, given that the passenger was over 40.
c) The passenger was male and over 40.

Question 23

Experimental studies utilize blinding to prevent researchers from biasing their measurements of the patients. But can they guess the treatment? In a study of a psychotherapeutic intervention, clinicians who were blinded were asked to guess what treatment their patients received. Data from that experiment are shown below. Assume that these data are indicative of the population of these treatments. ​
a) What is the probability that a correct guess is made for a randomly chosen subject?
b) If a randomly chosen subject received the placebo, what is the probability that the researcher correctly guesses the treatment?

Question 24

In which case the Law of Large Numbers cannot be applied?

A) When the number of trials is less than 1000.
B) When the events have the same likelihood of occurrence.
C) When the likelihood of an event is changing sufficiently from trial to trial.
D) When the real probability of an event occurring is not known.
E) When the number of probable events is greater than two.

Question 25

As every Girl Scout knows, statistics teachers seriously love Girl Scout Cookies. The number of boxes of GS cookies statistics teachers order, like all important decisions made by statistics teachers, is determined by independent rolls of a 4-sided fair die. If a one appears, 6 boxes are ordered; if any other number appears, 2 boxes are ordered. a) What is the probability that a statistics teacher places an order for 2 boxes of Girl Scout cookies? b) What is the probability that with two independently chosen statistics teachers will each order 6 boxes each? c) What is the probability that for two independently chosen statistics teachers the first will order 6 boxes and the second will order 2 boxes? d) What is the probability that for two independently chosen statistics teachers exactly one will order 6 boxes?

Question 26

Black bears (Ursus americanus) have a tendency to wander for food, and they have a high level of curiosity. These characteristics will sometimes get them into trouble when they travel through human-use areas such as parks. When they become "nuisances," the Park Service transplants them if possible to other areas. The outcomes of such transplants in Glacier National Park over a 10-year period are given in the table below: ​ a) From these data, estimate the probability that a randomly selected transplanted bear would be male and a nuisance in another area. b) From these data, estimate the probability that a randomly selected transplanted bear would be female or successfully transplanted. c) From these data, estimate the probability that a randomly selected transplanted bear would return to the capture area if she is a female. d) After combining the above data with other National Parks, officials estimated that only about 22% of black bears in all parks become enough of a nuisance to be transplanted. They further estimate that 84% of nuisance bears are male, and fifty percent of non-nuisance bears are females. If a randomly selected bear is observed to be a male, what is the probability it will be a nuisance?

Question 27

Select the correct statement concerning the Law of Total Probability and conditional probability.

A) The Law of Total Probability defines unconditional probability of an event P(E) using known conditional probabilities P(E|Bk) given that mutually exclusive events Bk have occurred, each Bk with a known probability itself. Conditional probability P(E|F) defines the probability of an event E given that an event F has occurred.
B) The Law of Total Probability defines unconditional probability of an event P(E) using known conditional probabilities P(E|Bk) given that mutually exclusive events Bk have occurred, each Bk with a known probability itself. Conditional probability P(E|F) defines the probability of an event F given that an event E has occurred.

Question 28

Students in two classes of upper-level mathematics were classified according to class standing and gender, resulting in the following table. Define events A, B, and C as follows: A = the event that the selected student is a female B = the event that the selected student is a male C = the event that the selected student is a senior. In the following table indicate by putting an X in the appropriate cells which is true for each pair of events. X = Yes, Blank = No

Question 29

Experimental studies utilize blinding to prevent researchers from biasing their measurements of the patients. But can they guess the treatment? In a study of a drug intervention, clinicians who were "blinded" were asked to guess what treatment their patients received. Data from that experiment are shown below. Assume that these data are indicative of the population of these treatments. ​ a) If a subject is selected at random, what is the probability that the researcher correctly guesses the treatment? b) If a randomly chosen subject received the standard treatment, what is the probability that the researcher correctly guesses the treatment?

Question 30

A small ferryboat transports vehicles from one island to another, and the order the vehicles arrive will lead to slightly different placement on the ferry. Consider the chance experiment where the type of vehicle--passenger (P) or recreational (R) vehicle--is recorded for each of the next two vehicles that arrive at the dock.
a) List all the events in the sample space.
b) Using your sample space in part (a), list the outcomes for each of the following events:
A = the event that both vehicles are passenger cars
B = the event that both vehicles are of the same type
C = the event that there is at least one passenger car A: B: C:
c) Display the sample space in the form of a tree diagram.

Question 31

Bayes' Rule allows to calculate the probability that a particular event was the reason that caused a specific test result after that test result is obtained.

Question 32

The basketball team at North Snowshoe High is in a little bit of trouble. Two of their players have just fouled out on technicals, and due to the flu the coach only has 4 players to choose from: three forwards and a guard who doubles as the team statistician. In his haste, the coach will randomly choose two of the four to go into the game, and of course will do so without replacement. What is the probability that the team statistician will be selected?

Question 33

In a special report, "Speed Demons," in the Cedar Rapids, IA, Gazette, reporters used a radar gun to check the speeds of motorists in two counties. They were shocked (!) to discover that some motorists were driving at a speed greater than the posted limits. They reported the following information: Suppose that one of these cars is selected at random. Consider the following events: F = the event that a driver is in a "fast lane" (speed limit above 25 mph). S = the event that a driver is speeding. ​ ​ b) From the information in part (a), calculate .
c) From the information in part (a), calculate .
d) From the information in part (a), calculate P(S).
e) These events are represented in a tree diagram below. What is the probability that a randomly selected driver will "follow the branches" to the X?

Question 34

In a few sentences, explain the difference between conditional probability and "ordinary" (unconditional) probability.