Question 1

A valid probability is any number between -1 and 1.

Accepted Answer

A valid probability must be between 0 and 1 inclusive. Probabilities cannot be negative, and they cannot be greater than 1.

Question 2

A study was conducted in a large population of adults concerning eyeglasses for correcting reading vision.Based on an examination by a qualified professional,the individuals were judged as to whether or not they needed to wear glasses for reading.In addition it was determined whether or not they were currently using glasses for reading.The following table provides the proportions found in the study.  What is the probability that a selected adult is judged to need eyeglasses but does not use them for reading?
A)0.42
B)0.18
C)0.54
D)0.60
E)0.36

Accepted Answer

The probability that a selected adult is judged to need eyeglasses is 0.7 (given in the table). The probability that the selected adult who needs eyeglasses does not use them for reading is 0.26 (also given in the table). Therefore, the probability that a selected adult is judged to need eyeglasses but does not use them for reading is the product of these two probabilities: 0.7 x 0.26 = 0.182 or approximately 0.18. So, the best choice is B.

Question 3

Ignoring twins and other multiple births,assume babies born at a hospital are independent events with the probability that a baby is a boy and the probability that a baby is a girl both equal to 0.5.Define event B = {at least one of the next two babies is a boy}.What is the probability of the complement of event B?
A)0.125
B)0.250
C)0.375
D)0.500

Accepted Answer

The complement of event B is that neither of the next two babies is a boy, which means both are girls. The probability of both babies being girls is 0.5 * 0.5 = 0.25.

Question 4

Suppose that A and B are two independent events.The probability that event A occurs is P(A)= 0.4 and that B occurs is P(B)= 0.2.What is the probability that A does not occur and that B also does not occur?
A)0.92
B)0.40
C)0.60
D)0.08
E)0.48

Accepted Answer

The probability that A does not occur is 1 - P(A) = 1 - 0.4 = 0.6, and the probability that B does not occur is 1 - P(B) = 1 - 0.2 = 0.8. Since A and B are independent, the probability that both A does not occur and B does not occur is the product of their individual probabilities: 0.6 * 0.8 = 0.48.

Question 5

A study was conducted in a large population of adults concerning eyeglasses for correcting reading vision.Based on an examination by a qualified professional,the individuals were judged as to whether or not they needed to wear glasses for reading.In addition it was determined whether or not they were currently using glasses for reading.The following table provides the proportions found in the study.  Suppose two adults are selected from the population independently and at random.What is the probability that both were judged to need eyeglasses and neither was using them for reading?
A)0.36
B)0.1296
C)0.0324
D)0.9216
E)0.18

Accepted Answer

Let A be the event that an adult is judged to need glasses and B be the event that an adult is not using glasses.

From the table, we know that:

P(A) = 0.5 (50% of adults were judged to need glasses)

P(B|A) = 0.4 (40% of adults judged to need glasses were not using them)

P(B|A') = 0.95 (95% of adults judged not to need glasses were not using them)

To find the probability that both adults meet the given criteria, we need to find the probability of the intersection of the two events:

P(A and B) = P(A) * P(B|A) * P(B|A')

= 0.5 * 0.4 * 0.95

= 0.19

Therefore, the probability that both adults were judged to need eyeglasses and neither was using them for reading is 0.19 or 0.0324 when rounded to four decimal places.

The correct answer is C.

Question 6

A study was conducted in a large population of adults concerning eyeglasses for correcting reading vision.Based on an examination by a qualified professional,the individuals were judged as to whether or not they needed to wear glasses for reading.In addition it was determined whether or not they were currently using glasses for reading.The following table provides the proportions found in the study.  If a single adult is selected at random from this large population,what is the probability that the adult is judged to need eyeglasses for reading?
A)0.46
B)0.42
C)0.78
D)0.60
E)0.40

Accepted Answer

The answer of A study was conducted in a large...

Question 7

If you draw an M&M candy at random from a bag of the candies,the candy you draw will have one of six colors.The probability of drawing each color depends on the proportion of each color among all candies made.Assume the table below gives the probabilities for the color of a randomly chosen M&M.  What is the probability of not drawing a red candy?
A)0.3
B)0.6
C)0.7
D)0.9

Accepted Answer

Since the table shows that the probability of drawing a red candy is 0.2, the probability of not drawing a red candy would be 1 minus the probability of drawing a red candy, which is 1 - 0.2 = 0.8. Therefore, the probability of not drawing a red candy is 0.8 or 0.7 when rounded to one decimal place. The best choice is C, 0.7, as it is the closest option to 0.8.

Question 8

At the end of a production run manufacturing rubber gaskets,items are sampled at random and inspected to determine if the item is acceptable (A)or defective (D).Suppose it is planned to select two items and determine if each is either A or D.Which set of outcomes describes the event that at least one of the items selected is defective?
A){AA,AD,DD}
B){AD,DA,DD}
C){DD}
D){AA,AD,DA}
E){AA,DD}

Accepted Answer

The event "at least one of the items selected is defective" can occur in three ways: selecting a defective item first and an acceptable item second (AD), selecting an acceptable item first and a defective item second (DA), or selecting two defective items (DD). Therefore, the set of outcomes that includes at least one of these three ways is B: {AD, DA, DD}. The other sets of outcomes do not include all three possibilities.

Question 9

Suppose a fair coin is flipped twice and the number of heads is counted.Which of the following is a valid probability model for the number of heads observed in two flips?
A) 
B) 
C) 
D)None of the above

Accepted Answer

The answer of Suppose a fair coin is flipped twice...

Question 10

Belgium has two official languages: French and Dutch.Assume that about 60% of the people speak Dutch and 40% of the people speak French.Define the event A as the event that two randomly selected Belgians speak the same language.What is the complement of event A?
A)0.48
B)0.52
C){Dutch,French}
D){(Dutch,French), (French,Dutch)}

Accepted Answer

The complement of event A is the event that two randomly selected Belgians speak different languages. This can happen in two ways: either the first person speaks Dutch and the second person speaks French, or the first person speaks French and the second person speaks Dutch. Therefore, the complement of event A is {(Dutch, French), (French, Dutch)}.

Question 11

A standard deck of cards has 52 cards.The cards have one of two colors: 26 cards in the deck are red and 26 are black.The cards have one of four denominations: 13 cards are hearts (red),13 cards are diamonds (red),13 cards are clubs (black),and 13 cards are spades (black).Two cards are selected at random.Event C is defined as the event that the first card is clubs,event R as the event that the first card is red,and event B as the event that the second card is black.Which events are disjoint?
A)R and B only
B)R and C only
C)R and B and R and C,but not B and C
D)None of the events are disjoint.

Accepted Answer

Events R and C are disjoint because a card cannot be both red and a club. However, events R and B are not disjoint because a red card can also be black (specifically, a red card of either hearts or diamonds). Similarly, events C and B are not disjoint because a club can also be red. Therefore, the only disjoint events are R and C.

Question 12

If you draw an M&M candy at random from a bag of the candies,the candy you draw will have one of six colors.The probability of drawing each color depends on the proportion of each color among all candies made.Assume the table below gives the probabilities for the color of a randomly chosen M&M.  What is the probability of drawing a yellow candy?
A)0.1
B)0.2
C)0.3
D)This is impossible to determine from the information given.

Accepted Answer

The answer of If you draw an M&M candy at...

Question 13

Ignoring twins and other multiple births,assume babies born at a hospital are independent events with the probability that a baby is a boy and the probability that a baby is a girl both equal to 0.5.Define events A = {the next two babies are boys} and B = {at least one of the next two babies is a boy}.What do we know about events A and B?
A)They are disjoint.
B)They are complements.
C)They are independent.
D)None of the above

Accepted Answer

Events A and B are not mutually exclusive, so they cannot be disjoint. They are also not complements, as the complement of event B would be "neither of the next two babies is a boy". Event A and Event B are not independent, since the occurrence of Event A affects the probability of Event B (if Event A occurs, then Event B also occurs). Therefore, the answer is D, "None of the above".

Question 14

If you draw an M&M candy at random from a bag of the candies,the candy you draw will have one of six colors.The probability of drawing each color depends on the proportion of each color among all candies made.Assume the table below gives the probabilities for the color of a randomly chosen M&M.  What is the probability that you draw neither a brown nor a green candy?
A)0.3
B)0.6
C)0.7
D)0.9

Accepted Answer

The answer of If you draw an M&M candy at...

Question 15

In a probability model,all possible outcomes together must have a probability of 1.

Accepted Answer

In a probability model, all possible outcomes combined should have a probability of 1. This means that when you add up the probability of all possible outcomes, the result should be 1. Therefore, the statement is true.

Question 16

Ignoring twins and other multiple births,assume babies born at a hospital are independent events with the probability that a baby is a boy and the probability that a baby is a girl both equal to 0.5.What is the probability that the next three babies are of the same sex?
A)0.125
B)0.250
C)0.375
D)0.500

Accepted Answer

The probability that all three babies are boys is (0.5)^3 = 0.125, and the probability that all three are girls is also (0.5)^3 = 0.125. Therefore, the probability that the next three babies are of the same sex is 0.125 + 0.125 = 0.250.

Question 17

If you draw an M&M candy at random from a bag of the candies,the candy you draw will have one of six colors.The probability of drawing each color depends on the proportion of each color among all candies made.Assume the table below gives the probabilities for the color of a randomly chosen M&M.  If you select two M&M's and the colors are independent,then what is the probability that both are the same color?
A)0.01
B)0.09
C)0.22
D)0.25

Accepted Answer

The answer of If you draw an M&M candy at...

Question 18

At the end of a production run manufacturing rubber gaskets,items are sampled at random and inspected to determine if the item is acceptable (A)or defective (D).Suppose it is planned to select two items and determine if each is either A or D.What is the sample space S of the outcomes?
A)S = {A or D}
B)S = {AA,DD}
C)S = {AA,AD,DA,DD}
D)This depends upon the assignment of probability to the outcomes of the sampling.
E)This can't be determined until the sample of two items has been collected.

Accepted Answer

The sample space S consists of all possible outcomes of selecting two items from the production run, where each item can either be acceptable (A) or defective (D). Therefore, each item has two possible outcomes, and there are a total of four possible outcomes for two items. The sample space S is {AA, AD, DA, DD}.

Question 19

Ignoring twins and other multiple births,assume babies born at a hospital are independent events with the probability that a baby is a boy and the probability that a baby is a girl both equal to 0.5.What is the probability that at least one of the next three babies is a boy?
A)0.125
B)0.333
C)0.750
D)0.875

Accepted Answer

The probability that all three babies are girls is (0.5)^3 = 0.125. Therefore, the probability that at least one of the three babies is a boy is 1 - 0.125 = 0.875.

Question 20

Suppose we roll a red die and a green die.Let R be the event that the number of spots showing on the red die is three or less,and G be the event that the number of spots showing on the green die is more than three.The events R and G are _________.
A)disjoint
B)complements
C)independent
D)reciprocals

Accepted Answer

The events R and G are independent because the outcome of the red die has no effect on the outcome of the green die. The events are not disjoint because it is possible for both events to occur at the same time (e.g. red die shows 1 and green die shows 6). The events are also not complements because they are not opposite outcomes (e.g. R does not necessarily imply not G and vice versa). The concept of reciprocals does not apply to events.

Quiz 4: Probability: the Study of Randomness

A valid probability is any number between -1 and 1.

Suppose that A and B are two independent events.The probability that event A occurs is P(A) = 0.4 and that B occurs is P(B) = 0.2.What is the probability that A does not occur and that B also does not occur?

Suppose a fair coin is flipped twice and the number of heads is counted.Which of the following is a valid probability model for the number of heads observed in two flips?

Belgium has two official languages: French and Dutch.Assume that about 60% of the people speak Dutch and 40% of the people speak French.Define the event A as the event that two randomly selected Belgians speak the same language.What is the complement of event A?

In a probability model,all possible outcomes together must have a probability of 1.

Ignoring twins and other multiple births,assume babies born at a hospital are independent events with the probability that a baby is a boy and the probability that a baby is a girl both equal to 0.5.What is the probability that the next three babies are of the same sex?

At the end of a production run manufacturing rubber gaskets,items are sampled at random and inspected to determine if the item is acceptable (A) or defective (D) .Suppose it is planned to select two items and determine if each is either A or D.What is the sample space S of the outcomes?

Ignoring twins and other multiple births,assume babies born at a hospital are independent events with the probability that a baby is a boy and the probability that a baby is a girl both equal to 0.5.What is the probability that at least one of the next three babies is a boy?

Suppose we roll a red die and a green die.Let R be the event that the number of spots showing on the red die is three or less,and G be the event that the number of spots showing on the green die is more than three.The events R and G are _________.