Question 1

According to the Mortgage Bankers Association,8% of U.S.mortgages were delinquent in 2011.A delinquent mortgage is one that has missed at least one payment but has not yet gone to foreclosure.A random sample of eight mortgages was selected.What is the probability that exactly one of these mortgages is delinquent?
A) 0.1296
B) 0.3570
C) 0.4926
D) 0.6157

Accepted Answer

This problem follows a binomial distribution with n=8 (number of mortgages) and p=0.08 (probability of a delinquent mortgage). We want to find the probability of exactly one delinquent mortgage, which is represented by the formula P(X=1) = (n choose x) * p^x * (1-p)^(n-x), where (n choose x) is the binomial coefficient. Plugging in the values, we get P(X=1) = (8 choose 1) * 0.08^1 * 0.92^7 = 0.3570, which is option B.

Question 2

Studies have shown that bats can consume an average of 10 mosquitoes per minute.What is the probability that a bat consumes four mosquitoes in a 30-second interval?
A) 0.1404
B) 0.1755
C) 0.3159
D) 0.0189

Accepted Answer

In a 30-second interval, a bat can consume an average of 5 mosquitoes (10 mosquitoes per minute divided by 2). We can use the Poisson distribution to calculate the probability of consuming exactly 4 mosquitoes in this interval:

λ = mean number of mosquito consumption = 5
x = number of mosquitoes of interest = 4

P(x=4) = e^(-λ) * λ^x / x!
P(x=4) = e^(-5) * 5^4 / 4!
P(x=4) ≈ 0.1755

Therefore, the probability that a bat consumes four mosquitoes in a 30-second interval is approximately 0.1755, which corresponds to choice B.

Question 3

According to the Mortgage Bankers Association,8% of U.S.mortgages were delinquent in 2011.A delinquent mortgage is one that has missed at least one payment but has not yet gone to foreclosure.A random sample of eight mortgages was selected.What is the probability that fewer than three of these mortgages are delinquent?
A) 0.7018
B) 0.7583
C) 0.9789
D) 0.8225

Accepted Answer

The probability of a mortgage being delinquent is 0.08. The probability of a mortgage not being delinquent is 1 - 0.08 = 0.92. The probability of fewer than three mortgages being delinquent is the sum of the probabilities of 0, 1, and 2 mortgages being delinquent. The probability of 0 mortgages being delinquent is 0.92^8 = 0.4305. The probability of 1 mortgage being delinquent is 8 * 0.08 * 0.92^7 = 0.3020. The probability of 2 mortgages being delinquent is 28 * 0.08^2 * 0.92^6 = 0.1644. The sum of these probabilities is 0.4305 + 0.3020 + 0.1644 = 0.9789.

Question 4

An urn contains 12 balls,five of which are red.Selection of a red ball is desired and is therefore considered to be a success.If three balls are selected,what is the expected value of the distribution of the number of selected red balls?
A) 0.4167
B) 0.8333
C) 0.5833
D) 1.2500

Accepted Answer

The answer of An urn contains 12 balls,five of which...

Question 5

The following table shows the number of students who did not attend statistics class over the 30 class meetings last semester.  What is the expected number of absent students per class for the next semester?
A) 2.123
B) 1.800
C) 1.965
D) 1.246

Accepted Answer

The expected value of the number of absent students per class can be calculated by taking the average of the number of absent students over the 30 class meetings. Therefore, (87+57+31+18+20+25+35+28+26+31)/30 = 1.8. Therefore, the best choice is B.

Question 6

Forty-four percent of consumers with credit cards carry balances from month to month.Four consumers with credit cards are randomly selected.What is the probability that all consumers carry a credit card balance?
A) 0.1015
B) 0.0375
C) 0.3133
D) 0.1390

Accepted Answer

The probability that one randomly selected consumer carries a credit card balance is 44%. So, the probability that all four consumers carry a credit card balance is:
P(all carry balance) = 0.44^4 = 0.0375 (rounded to four decimal places)
Therefore, the correct answer is B.

Question 7

Forty-four percent of consumers with credit cards carry balances from month to month.Four consumers with credit cards are randomly selected.What is the probability that fewer than two consumers carry a credit card balance?
A) 0.4074
B) 0.6734
C) 0.7717
D) 0.3643

Accepted Answer

We can use the binomial distribution to solve this problem, where carrying a credit card balance is a success (with probability 0.44) and not carrying a credit card balance is a failure (with probability 0.56). Let X be the number of consumers out of four who carry a credit card balance. We want to find P(X < 2). P(X < 2) = P(X = 0) + P(X = 1) Using the binomial probability formula: P(X = k) = (4 choose k) * 0.44^k * 0.56^(4-k) where (4 choose k) = 4! / (k! * (4-k)!) Therefore, P(X = 0) = (4 choose 0) * 0.44^0 * 0.56^4 = 0.1308 P(X = 1) = (4 choose 1) * 0.44^1 * 0.56^3 = 0.3843 Therefore, P(X < 2) = 0.1308 + 0.3843 = 0.5151 But the problem asks for the probability of fewer than two consumers carrying a balance, therefore, we can calculate P(X < 2) = P(X = 0) + P(X = 1) which is P(X ≤ 1) P(X ≤ 1) = 1 - P(X > 1) = 1 - P(X >= 2) Now, calculating probability of P(X >= 2) P(X >= 2) = P(X = 2) + P(X = 3) + P(X = 4) where, P(X = 2) = (4 choose 2) * 0.44^2 * 0.56^2 = 0.3455 P(X = 3) = (4 choose 3) * 0.44^3 * 0.56^1 = 0.1899 P(X = 4) = (4 choose 4) * 0.44^4 * 0.56^0 = 0.0259 Therefore, P(X >= 2) = 0.3455 + 0.1899 + 0.0259 = 0.5613 So, P(X < 2) = 1 - P(X >= 2) = 1 - 0.5613 = 0.4387 P(X < 2) = 0.4387 which is closest to 0.4074 in the options given, therefore our answer will be option A.

Question 8

According to the Department of Transportation,27% of domestic flights were delayed in 2007 at JFK airport.Five flights are randomly selected at JFK.What is the probability that all five flights are delayed?
A) 0.0194
B) 0.0208
C) 0.0014
D) 0.0222

Accepted Answer

The probability of one flight being delayed is 0.27. Since each flight's delay is independent, we can calculate the probability of all five flights being delayed by multiplying the probabilities of each flight being delayed: 
0.27 x 0.27 x 0.27 x 0.27 x 0.27 = 0.0014 or 0.14%. Therefore, the best choice is C.

Question 9

The following table shows the number of students who did not attend statistics class over the 30 class meetings last semester.  What is the probability that there will be more than one student absent in class today?
A) 0.54
B) 0.87
C) 0.70
D) 0.63

Accepted Answer

The answer of The following table shows the number of...

Question 10

A professor has learned that three students in his class of 20 will cheat on the final exam.He decides to focus his attention on four randomly chosen students during the exam.What is the probability that he finds at least one of the students cheating?
A) 0.4912
B) 0.4211
C) 0.5088
D) 0.5789

Accepted Answer

The probability that a randomly chosen student will cheat is 3/20. Therefore, the probability that a randomly chosen student will not cheat is 17/20. The probability that all four of the students chosen by the professor will not cheat is (17/20)^4 = 0.456. Therefore, the probability that at least one of the students will cheat is 1 - 0.456 = 0.544. However, the question asks for the probability that the professor finds at least one of the students cheating, not the probability that at least one of the students cheats. It is possible that a student will cheat but the professor will not catch them. Therefore, the probability that the professor finds at least one of the students cheating is lower than 0.544, but higher than 0.456. The only choice that falls within this range is C, 0.5088.

Question 11

Suppose a baseball team has 14 players on the roster who are not members of the pitching staff.Of those 14 players,assume that three have recently taken a performance-enhancing drug.Suppose the league decides to randomly test five members of the team.What is the probability that exactly two of the tested players are found to have taken a performance-enhancing drug?
A) 0.2308
B) 0.2473
C) 0.4945
D) 0.7692

Accepted Answer

The probability that exactly two out of the five tested players have taken a performance-enhancing drug can be calculated using the binomial probability formula:
P(X = 2) = (C(3,2) * C(11,3)) / C(14,5)
where C(n,r) represents the number of combinations of n items taken r at a time.

Simplifying this expression gives:
P(X = 2) = (3 * 165) / 2002
P(X = 2) = 495 / 2002
P(X = 2) ≈ 0.2473

Therefore, the best choice is B, with a probability of 0.2473.

Question 12

A six-sided,unfair (weighted)die has the following probability distribution.  Find the probability of rolling a 3 or less.

Accepted Answer

The answer of A six-sided,unfair (weighted)die has the following probability...

Question 13

Suppose a baseball team has 14 players on the roster who are not members of the pitching staff.Of those 14 players,assume that three have recently taken a performance-enhancing drug.Suppose the league decides to randomly test five members of the team.What is the probability that at least one of the tested players is found to have taken a performance-enhancing drug?
A) 0.2308
B) 0.2473
C) 0.4945
D) 0.7692

Accepted Answer

The probability of at least one player being found to have taken a performance-enhancing drug is easier to calculate by finding the complement, which is the probability that none of the tested players have taken the drug, and then subtracting that from 1. There are 11 players who haven't taken the drug, so the probability of selecting 5 of these from the total 14 without selecting any of the 3 who did take the drug is (11 choose 5) / (14 choose 5). This gives 1 - (462/600) = 1 - 0.77 = 0.2308 for the complement, so the probability of at least one being found is 1 - 0.2308 = 0.7692.

Question 14

An urn contains 12 balls,five of which are red.The selection of a red ball is desired and is therefore considered to be a success.If a person draws three balls from the urn,what is the probability of two successes?
A) 0.1591
B) 0.3182
C) 0.6810
D) 0.8409

Accepted Answer

We can use the binomial probability formula to compute the probability of exactly 2 successes in 3 draws: P(X=2) = (3 choose 2) * (5/12)^2 * (7/12)^1 = 3 * 25/144 * 7/12 = 0.3182. Therefore, the correct answer is choice B.

Question 15

Which of the following is true about the hypergeometric distribution?
A) The trials are independent and the probability of success may change from trial to trial.
B) The trials are independent and the probability of success does not change from trial to trial.
C) The trials are not independent and the probability of success may change from trial to trial.
D) The trials are not independent and the probability of success does not change from trial to trial.

Accepted Answer

The hypergeometric distribution is used when the sample size is much smaller than the population size, and the trials are not independent (as with the binomial distribution). Also, the probability of success can change from trial to trial as the sample is drawn without replacement from the population. Therefore, choice C is the correct answer.

Question 16

According to geologists,the San Francisco Bay Area experiences five earthquakes with a magnitude of 6.5 or greater every 100 years.What is the standard deviation of the number of earthquakes with a magnitude of 6.5 or greater striking the San Francisco Bay Area in the next 40 years?
A) 1.414
B) 2.000
C) 2.236
D) 5.000

Accepted Answer

The answer of According to geologists,the San Francisco Bay Area...

Question 17

There are currently 18 pit bulls at the pound.Of the 18 pit bulls,four have attacked another dog in the last year.Joe,a member of the staff,randomly selects six of the pit bulls for his group.What is the probability that at least one of the pit bulls in Joe's group attacked another dog last year?
A) 0.1618
B) 0.4314
C) 0.5686
D) 0.8382

Accepted Answer

The probability that none of the six pit bulls selected by Joe have attacked another dog is $\frac{\binom{14}{6}}{\binom{18}{6}} = \frac{3003}{18564}$. Therefore, the probability that at least one of the six pit bulls selected by Joe have attacked another dog is $1 - \frac{3003}{18564} = 0.8382$, which is answer choice $\boxed{	extbf{(D)}}$.

Question 18

According to the Department of Transportation,27% of domestic flights were delayed in 2007 at JFK airport.Five flights are randomly selected at JFK.What is the probability that all five flights are on time?
A) 0.1513
B) 0.7927
C) 0.8487
D) 0.2073

Accepted Answer

The answer of According to the Department of Transportation,27% of...

Question 19

There are currently 18 pit bulls at the pound.Of the 18 pit bulls,four have attacked another dog in the last year.Joe,a member of the staff,randomly selects six of the pit bulls for his group.What is the probability that exactly one of the pit bulls in Joe's group attacked another dog last year?
A) 0.1618
B) 0.3235
C) 0.4314
D) 0.4853

Accepted Answer

To find the probability that exactly one of the pit bulls in Joe's group attacked another dog last year, we need to use the binomial distribution. Letting $p$ be the probability that any one of the pit bulls in the pound attacked another dog last year, we have $p=\frac{4}{18}=\frac{2}{9}$. The probability of exactly one success (i.e. one pit bull that attacked another dog) in six trials (i.e. six pit bulls selected for Joe's group) is given by the binomial probability:
$$P(X=1) = \binom{6}{1}p^1(1-p)^5 = 6\left(\frac{2}{9}ight)\left(\frac{7}{9}ight)^5 \approx 0.4314$$
Therefore, the best choice is $\boxed{	extbf{(C)}}$.

Question 20

Studies have shown that bats can consume an average of 10 mosquitoes per minute.What is the probability that a bat does not consume any mosquitoes in a 30-second interval?
A) 0.0067
B) 0.1755
C) 0.0023
D) 0.0090

Accepted Answer

The answer of Studies have shown that bats can consume...

Quiz 5: Discrete Probability Distributions

Studies have shown that bats can consume an average of 10 mosquitoes per minute.What is the probability that a bat consumes four mosquitoes in a 30-second interval?

An urn contains 12 balls,five of which are red.Selection of a red ball is desired and is therefore considered to be a success.If three balls are selected,what is the expected value of the distribution of the number of selected red balls?

The following table shows the number of students who did not attend statistics class over the 30 class meetings last semester. What is the expected number of absent students per class for the next semester?

Forty-four percent of consumers with credit cards carry balances from month to month.Four consumers with credit cards are randomly selected.What is the probability that all consumers carry a credit card balance?

Forty-four percent of consumers with credit cards carry balances from month to month.Four consumers with credit cards are randomly selected.What is the probability that fewer than two consumers carry a credit card balance?

According to the Department of Transportation,27% of domestic flights were delayed in 2007 at JFK airport.Five flights are randomly selected at JFK.What is the probability that all five flights are delayed?

The following table shows the number of students who did not attend statistics class over the 30 class meetings last semester. What is the probability that there will be more than one student absent in class today?

A professor has learned that three students in his class of 20 will cheat on the final exam.He decides to focus his attention on four randomly chosen students during the exam.What is the probability that he finds at least one of the students cheating?

A six-sided,unfair (weighted)die has the following probability distribution. Find the probability of rolling a 3 or less.

An urn contains 12 balls,five of which are red.The selection of a red ball is desired and is therefore considered to be a success.If a person draws three balls from the urn,what is the probability of two successes?

Which of the following is true about the hypergeometric distribution?

According to geologists,the San Francisco Bay Area experiences five earthquakes with a magnitude of 6.5 or greater every 100 years.What is the standard deviation of the number of earthquakes with a magnitude of 6.5 or greater striking the San Francisco Bay Area in the next 40 years?

There are currently 18 pit bulls at the pound.Of the 18 pit bulls,four have attacked another dog in the last year.Joe,a member of the staff,randomly selects six of the pit bulls for his group.What is the probability that at least one of the pit bulls in Joe's group attacked another dog last year?

According to the Department of Transportation,27% of domestic flights were delayed in 2007 at JFK airport.Five flights are randomly selected at JFK.What is the probability that all five flights are on time?

There are currently 18 pit bulls at the pound.Of the 18 pit bulls,four have attacked another dog in the last year.Joe,a member of the staff,randomly selects six of the pit bulls for his group.What is the probability that exactly one of the pit bulls in Joe's group attacked another dog last year?

Studies have shown that bats can consume an average of 10 mosquitoes per minute.What is the probability that a bat does not consume any mosquitoes in a 30-second interval?