Question 1

A coin is tossed four times. Let the random variable X denote the number of heads that occur. Let H denotes the outcome "heads" and T denotes the outcome "tails". Find the value assigned to each outcome of the experiment by the random variable X. ​ 
A) 
B) 
C)  
D)

Accepted Answer

The value assigned to each outcome of the experiment by the random variable X is:
- X(HHHH) = 4
- X(HHHT) = 3
- X(HHTH) = 3
- X(HHTT) = 2
- X(HTHH) = 3
- X(HTHT) = 2
- X(HTTH) = 2
- X(HTTT) = 1
- X(THHH) = 3
- X(THHT) = 2
- X(THTH) = 2
- X(THTT) = 1
- X(TTHH) = 2
- X(TTHT) = 1
- X(TTTH) = 1
- X(TTTT) = 0

Option B correctly lists the values assigned to each outcome.

Question 2

An examination consisting of ten true-or-false questions was taken by a class of 100 students. The probability distribution of the random variable X, where X denotes the number of questions answered correctly by a randomly chosen student, is represented by the accompanying histogram. The rectangle with base centered on the number 8 is missing. What should be the height of this rectangle? ​  ​
A) 0.35
B) 0.25
C) 0.4
D) 0.2

Accepted Answer

A

Question 3

A die is cast repeatedly until a 4 falls uppermost. Let the random variable X denote the number of times the die is cast. What are the values that X may assume? ​
A) X may assume the values in the set S = {1, 2, 3, ...}
B) X may assume the values in the set S = {4}
C) X may assume the values in the set S = {1, 2, 3, 4, 5, 6}

Accepted Answer

X represents the number of trials until the first success (rolling a 4), which can theoretically take any number of trials from 1 to infinity, hence the set {1, 2, 3, ...}.

Question 4

An examination consisting of ten true-or-false questions was taken by a class of 100 students. The probability distribution of the random variable X, where X denotes the number of questions answered correctly by a randomly chosen student, is represented by the accompanying histogram. The rectangle with base centered on the number 8 is missing. What should be the height of this rectangle? ​  ​
A) 0.15
B) 0.3
C) 0.25
D) 0.1

Accepted Answer

The answer of An examination consisting of ten true-or-false questions...

Question 5

Give the range of values that the random variable X may assume and classify the random variable as finite discrete, infinite discrete, or continuous. ​
X = The distance a commuter travels to work
​
A)  . The random variable is infinite discrete.
B)  . The random variable is continuous.
C)  . The random variable is finite discrete.

Accepted Answer

The distance a commuter travels to work can take on any real value between 0 and infinity, making it a continuous random variable.

Question 6

Three balls are selected at random without replacement from an urn containing four white balls and six red balls. Let the random variable X denote the number of white balls drawn. List the outcomes of the experiment.
A) WWW, WWR, WRW, RWW, WRR, RWR, RRW
B) WWW, WWR, WRW, RWW
C) WWW, WWR, WRW, RWW, WRR, RWR, RRW, RRR
D) WRR, RWR, RRW, RRR

Accepted Answer

The outcomes include all possible combinations of drawing three balls from the urn, which can range from drawing all white balls (WWW) to all red balls (RRR), including mixed outcomes.

Question 7

The probability distribution of the random variable X is shown in the accompanying table:  Find  ,  and  .
A)   
B)   
C)   
D)

Accepted Answer

The answer of The probability distribution of the random variable...

Question 8

The probability distribution of the random variable X is shown in the accompanying table. 
A) 0.35
B) 0.6
C) 0.55
D) 0.45
E) 0.3

Accepted Answer

The answer of The probability distribution of the random variable...

Question 9

Cards are selected one at a time without replacement from a well-shuffled deck of 52 cards until an ace is drawn. Let X denote the random variable that gives the number of cards drawn. What values may X assume? ​
A) X may assume the values in the set S = {1, 2, ...,52}
B) X may assume the values in the set S = {1, 2, ...,47}
C) X may assume the values in the set S = {1, 2, ...,48}
D) X may assume the values in the set S = {1, 2, ...,49}

Accepted Answer

Since there are four aces in a standard deck of 52 cards, the first ace can appear on the first card drawn or the 48th card drawn (if it hasn't appeared in the first 47 cards drawn). Therefore, X can take on values in the set S = {1, 2, ..., 49} and the best choice is D).

Question 10

Let X denote the random variable that gives the sum of the faces that fall uppermost when two fair dice are cast. Find P ( X = 5 ). ​
A) 
B) 
C) 
D)

Accepted Answer

The answer of Let X denote the random variable that...

Question 11

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The area of a histogram associated with a probability distribution is a number between 0 and 1. 
A) False. The area is exactly equal to one.
B) True. This follows from the definition.

Accepted Answer

The area under the histogram of a probability distribution represents the total probability of all outcomes, which must sum to 1 by the definition of a probability distribution.

Question 12

Three cards are drawn from a well-shuffled deck of 52 playing cards. Let X denote the number of aces drawn. Find  . ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Three cards are drawn from a well-shuffled...

Question 13

Give the range of values that the random variable X may assume and classify the random variable as finite discrete, infinite discrete, or continuous. ​
X = The number of hours a child watches television on a given day
​
A)  , Finite Continuous
B) Any positive integer, Infinite discrete
C)  , discrete
D)  , Finite discrete
E)  , Continuous

Accepted Answer

The answer of Give the range of values that the...

Question 14

Give the range of values that the random variable X may assume and classify the random variable as finite discrete, infinite discrete, or continuous. X = The number of times an accountant takes the CPA examination before passing
A) Any integer; infinite discrete
B) 0 ≤ x < ∞; continuous
C) Any positive integer; infinite discrete
D) 0 ≤ x < 24; continuous
E) Any positive integer; infinite continuous

Accepted Answer

The random variable X can only take positive integer values (1, 2, 3, etc.) as an accountant has to take the CPA exam at least once before passing it. Since there is no upper limit on the number of times one can take the exam, X can take infinitely many values. Thus, the random variable is infinite discrete, and the only choice that reflects these characteristics is C.

Question 15

Give the range of values that the random variable X may assume and classify the random variable as finite discrete, infinite discrete, or continuous. ​
X = The number of boys in a three-child family
​
A) {1, 2, 3}. The variable is finite discrete.
B) {1, 2, 3}. The variable is continuous.
C) {1, 2, 3}. The variable is infinite discrete.

Accepted Answer

The range of values that X may assume is {1, 2, 3}, since there can only be 0, 1, 2, or 3 boys in a three-child family. The variable is finite discrete because it can only take on a finite number of possible values, and those values are integers.

Question 16

Give the range of values that the random variable X may assume and classify the random variable as finite discrete, infinite discrete, or continuous. ​
X = The number of times a die is thrown until a 3 appears.
​
A) X may assume the values of any positive integer. The random variable is finite discrete.
B) X may assume the values of any positive integer. The random variable is continuous.
C) X may assume the values in the set S = {1, 2, 3, 4, 5, 6}. The random variable is finite discrete.
D) X may assume the values of any positive integer. The random variable is infinite discrete.

Accepted Answer

The random variable X can take any positive integer value, so it is not limited to a finite set of numbers. Therefore, it is not finite discrete. Additionally, X only takes on integer values, so it is not continuous. X is therefore an infinite discrete random variable. The correct choice is D.

Question 17

An examination consisting of ten true-or-false questions was taken by a class of 100 students. The probability distribution of the random variable X, where X denotes the number of questions answered correctly by a randomly chosen student, is represented by the accompanying histogram. The rectangle with base centered on the number 8 is missing. What should be the height of this rectangle? ​  ​
A) 0.3
B) 0.35
C) 0.45
D) 0.5

Accepted Answer

The answer of An examination consisting of ten true-or-false questions...

Question 18

Find the condition on the numbers a and b such that the table below gives the probability distribution of the random variable X.  ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the condition on the numbers a...

Question 19

Give the range of values that the random variable X may assume and classify the random variable as finite discrete, infinite discrete, or continuous. ​
X = The number of defective watches in a sample of four watches.
​
A) {0,1,2,3,4}; The random variable is finite discrete
B) {0,1,2,3,4}; The random variable is infinite discrete
C) X may assume the values of any positive integer. The random variable is infinite discrete.
D) X may assume the values of any positive integer. The random variable is continuous.

Accepted Answer

The range of values that X may assume are {0,1,2,3,4}, which is a finite set. Therefore, the random variable is finite discrete.

Question 20

Determine whether the table gives the probability distribution of the random variable X. Explain your answer. 
A) No, the sum of the probability assigned to the value of the random variable X is greater than 1.
B) No, the sum of the probability assigned to the value of the random variable X is not equal to 1.
C) No, the probability assigned to a value of the random variable X cannot be negative.
D) Yes, the sum of the probability assigned to the value of the random variable X is equal to 1.
E) No, the sum of the probability assigned to the value of the random variable X is less than 1.

Accepted Answer

The answer of Determine whether the table gives the probability...

Quiz 8: Probability Distributions and Statistics

A coin is tossed four times. Let the random variable X denote the number of heads that occur. Let H denotes the outcome "heads" and T denotes the outcome "tails". Find the value assigned to each outcome of the experiment by the random variable X.

A die is cast repeatedly until a 4 falls uppermost. Let the random variable X denote the number of times the die is cast. What are the values that X may assume?

Give the range of values that the random variable X may assume and classify the random variable as finite discrete, infinite discrete, or continuous. X = The distance a commuter travels to work

Three balls are selected at random without replacement from an urn containing four white balls and six red balls. Let the random variable X denote the number of white balls drawn. List the outcomes of the experiment.

The probability distribution of the random variable X is shown in the accompanying table: Find , and .

The probability distribution of the random variable X is shown in the accompanying table.

Cards are selected one at a time without replacement from a well-shuffled deck of 52 cards until an ace is drawn. Let X denote the random variable that gives the number of cards drawn. What values may X assume?

Let X denote the random variable that gives the sum of the faces that fall uppermost when two fair dice are cast. Find P ( X = 5 ) .

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The area of a histogram associated with a probability distribution is a number between 0 and 1.

Three cards are drawn from a well-shuffled deck of 52 playing cards. Let X denote the number of aces drawn. Find .

Give the range of values that the random variable X may assume and classify the random variable as finite discrete, infinite discrete, or continuous. X = The number of hours a child watches television on a given day

Give the range of values that the random variable X may assume and classify the random variable as finite discrete, infinite discrete, or continuous. X = The number of times an accountant takes the CPA examination before passing

Give the range of values that the random variable X may assume and classify the random variable as finite discrete, infinite discrete, or continuous. X = The number of boys in a three-child family

Give the range of values that the random variable X may assume and classify the random variable as finite discrete, infinite discrete, or continuous. X = The number of times a die is thrown until a 3 appears.

Find the condition on the numbers a and b such that the table below gives the probability distribution of the random variable X.

Give the range of values that the random variable X may assume and classify the random variable as finite discrete, infinite discrete, or continuous. X = The number of defective watches in a sample of four watches.

Determine whether the table gives the probability distribution of the random variable X. Explain your answer.

Quiz 8: Probability Distributions and Statistics

A coin is tossed four times. Let the random variable X denote the number of heads that occur. Let H denotes the outcome "heads" and T denotes the outcome "tails". Find the value assigned to each outcome of the experiment by the random variable X. ​

A die is cast repeatedly until a 4 falls uppermost. Let the random variable X denote the number of times the die is cast. What are the values that X may assume? ​

Give the range of values that the random variable X may assume and classify the random variable as finite discrete, infinite discrete, or continuous. ​ X = The distance a commuter travels to work ​

Three balls are selected at random without replacement from an urn containing four white balls and six red balls. Let the random variable X denote the number of white balls drawn. List the outcomes of the experiment.

The probability distribution of the random variable X is shown in the accompanying table: Find , and .

The probability distribution of the random variable X is shown in the accompanying table.

Cards are selected one at a time without replacement from a well-shuffled deck of 52 cards until an ace is drawn. Let X denote the random variable that gives the number of cards drawn. What values may X assume? ​

Let X denote the random variable that gives the sum of the faces that fall uppermost when two fair dice are cast. Find P ( X = 5 ) . ​