Question 1

A quiz consists of 920 true or false questions. If the student guesses on each question, what is the mean numberof correct answers?

Accepted Answer

Since each question has a 50% chance of being answered correctly when guessing, the mean number of correct answers out of 920 questions is 920 * 0.5 = 460.

Question 2

A recent survey found that 69% of all adults over 50 wear sunglasses for driving. In a random sample of 20adults over 50, what is the mean and standard deviation of those that wear sunglasses?

Accepted Answer

The mean of those that wear sunglasses is found by multiplying the probability by the sample size: $0.69 \times 20 = 13.8$. The standard deviation is calculated using the formula for the standard deviation of a binomial distribution, $\sqrt{n \times p \times (1-p)}$, where $n$ is the sample size, $p$ is the probability of success (wearing sunglasses), and $1-p$ is the probability of failure. Substituting the given values: $\sqrt{20 \times 0.69 \times (1-0.69)} = 2.06833266$.

Question 3

A history professor decides to give a 12-question true-false quiz. She wants to choose the passing grade suchthat the probability of passing a student who guesses on every question is less than 0.10. What score should beset as the lowest passing grade?

Accepted Answer

The probability of guessing correctly on a true-false question is 0.5. Using the binomial distribution, we calculate the probability of getting at least 9 correct answers out of 12 by guessing. The cumulative probability of getting 9, 10, 11, or 12 correct answers is less than 0.10, making 9 the lowest passing grade.

Question 4

According to government data, the probability that an adult was never in a museum is 15%. In a randomsurvey of 10 adults, what is the probability that two or fewer were never in a museum?

Accepted Answer

This is a binomial probability problem where n = 10 (number of trials), p = 0.15 (probability of success, which in this context is an adult never having been in a museum), and we are looking for the probability of x ≤ 2 successes (two or fewer adults never having been in a museum). The formula for binomial probability is P(X = x) = (nCx) * p^x * (1-p)^(n-x), where nCx is the combination of n items taken x at a time. To find the probability of two or fewer, we calculate P(X = 0), P(X = 1), and P(X = 2) and sum these probabilities. The exact calculations can be tedious without a calculator but generally result in a sum that is closest to the option A) 0.820, indicating a high likelihood that in a random survey of 10 adults, two or fewer will have never been in a museum.

Question 5

According to insurance records, a car with a certain protection system will be recovered 92% of the time. If 800stolen cars are randomly selected, what is the mean and standard deviation of the number of cars recoveredafter being stolen?

Accepted Answer

The mean of the number of cars recovered is found by multiplying the total number of stolen cars by the probability of recovery: 800 * 0.92 = 736. The standard deviation is calculated using the formula for the standard deviation of a binomial distribution, which is sqrt(n*p*(1-p)), where n is the number of trials, p is the probability of success on each trial. Thus, the standard deviation is sqrt(800*0.92*0.08) = 7.67333044.

Question 6

According to government data, the probability that an adult was never in a museum is 15%. In a randomsurvey of 10 adults, what is the probability that at least eight were in a museum?

Accepted Answer

The probability that an adult was in a museum is 1 - 0.15 = 0.85. To find the probability that at least 8 out of 10 adults were in a museum, we calculate the probabilities for exactly 8, 9, and 10 adults being in a museum and sum these probabilities. This can be done using the binomial probability formula, which leads to a cumulative probability that matches choice A.

Question 7

In a recent survey, 80% of the community favored building a health center in their neighborhood. If 15 citizensare chosen, what is the mean number favoring the health center?

Accepted Answer

The mean number favoring the health center can be found by multiplying the total number of citizens chosen by the percentage that favored the health center. Thus, 15 citizens * 80% = 12 citizens favor the health center.

Question 8

A quiz consists of 780 true or false questions. If the student guesses on each question, what is the standarddeviation of the number of correct answers?

Accepted Answer

The standard deviation for a binomial distribution is calculated using the formula √(n*p*(1-p)), where n is the number of trials (780) and p is the probability of success (0.5 for guessing). This results in √(780*0.5*0.5) = √195 = 13.96424.

Question 9

We believe that 86% of the population of all Calculus I students consider calculus an exciting subject. Supposewe randomly and independently selected 29 students from the population. If the true percentage is really 86%,find the probability of observing 28 or more of the students who consider calculus to be an exciting subject inour sample of 29.

Accepted Answer

This problem can be solved using the binomial probability formula, which is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successful trials, and p is the probability of success on a single trial. Here, n = 29, p = 0.86, and we are interested in k = 28 and k = 29. The probability of observing exactly 28 students who consider calculus exciting is calculated using the binomial formula, and similarly for 29 students. The total probability of observing 28 or more students is the sum of these two probabilities. The correct answer is obtained by performing these calculations, which results in a probability of approximately 0.072098.

Question 10

A quiz consists of 10 true or false questions. To pass the quiz a student must answer at least eight questionscorrectly. If the student guesses on each question, what is the probability that the student will pass the quiz?

Accepted Answer

The probability of guessing each question correctly is 0.5. To find the probability of passing (getting at least 8 out of 10 questions right), we calculate the probabilities of getting exactly 8, 9, and 10 questions right and sum them up. Using the binomial probability formula, P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where n=10, p=0.5, k=8, 9, 10, and C(n, k) is the combination of n items taken k at a time, we get the sum of these probabilities as approximately 0.055.

Question 11

A recent article in the paper claims that government ethics are at an all-time low. Reporting on a recent sample,the paper claims that 45% of all constituents believe their representative possesses low ethical standards.Suppose 20 of a representativeʹs constituents are randomly and independently sampled. Assuming the paperʹsclaim is correct, find the probability that more than eight but fewer than 12 of the 20 constituents sampledbelieve their representative possesses low ethical standards.

Accepted Answer

The probability that more than eight but fewer than 12 constituents believe their representative possesses low ethical standards can be calculated using the binomial probability formula. Given n = 20, p = 0.45, we calculate P(9 ≤ X ≤ 11) which results in approximately 0.454929.

Question 12

A recent article in the paper claims that government ethics are at an all-time low. Reporting on a recent sample,the paper claims that 38% of all constituents believe their representative possesses low ethical standards.Assume that responses were randomly and independently collected. A representative of a district with 1,000people does not believe the paperʹs claim applies to her. If the claim is true, how many of the representativeʹsconstituents believe the representative possesses low ethical standards?

Accepted Answer

If 38% of all constituents believe their representative possesses low ethical standards, then in a district with 1,000 people, 38% of 1,000 would believe so. This is calculated as 0.38 * 1000 = 380.

Question 13

A psychic network received telephone calls last year from over 1.5 million people. A recent article attempts toshed some light onto the credibility of the psychic network. One of the psychic networkʹs psychics agreed totake part in the following experiment. Five different cards are shuffled, and one is chosen at random. Thepsychic will then try to identify which card was drawn without seeing it. Assume that the experiment wasrepeated 50 times and that the results of any two experiments are independent of one another. If we assumethat the psychic is a fake (i.e., they are merely guessing at the cards and have no psychic powers), find theprobability that they guess at least three correctly.

Accepted Answer

The probability of guessing a card correctly is 1/5, and incorrectly is 4/5. To find the probability of guessing at least 3 cards correctly out of 50, we use the binomial distribution formula. However, calculating the probability directly for 3 to 50 correct guesses is cumbersome. Instead, it's easier to calculate the probability of guessing 0, 1, or 2 cards correctly and subtracting from 1. The result is very close to 1, indicating a high probability of guessing at least 3 cards correctly by chance, which aligns with choice A.

Question 14

According to the Federal Communications Commission, 70% of all U.S. households have vcrs. In a randomsample of 15 households, what is the probability that at least 13 have vcrs?

Accepted Answer

This is a binomial probability problem where n = 15 (number of trials), p = 0.7 (probability of success), and we are looking for the probability of at least 13 successes (x ≥ 13). This can be calculated using the binomial probability formula or a binomial distribution table. The calculation involves finding the probability of exactly 13 households having VCRs, exactly 14 households having VCRs, and all 15 households having VCRs, and then summing these probabilities. The correct answer, after performing these calculations, is 0.1268.

Question 15

According to the Federal Communications Commission, 70% of all U.S. households have vcrs. In a randomsample of 15 households, what is the probability that the number of households with vcrs is between 10 and 12,inclusive?

Accepted Answer

The probability of a household having a VCR is 0.7. To find the probability of having between 10 and 12 households (inclusive) with VCRs out of 15, we use the binomial probability formula. Calculating the probabilities for exactly 10, 11, and 12 households and summing them gives us the total probability, which matches choice A, 0.5947.

Question 16

Assume that male and female births are equally likely and that the birth of any child does not affect theprobability of the gender of any other children. Suppose that 200 couples each have a baby; find the mean andstandard deviation for the number of boys in the 200 babies.

Accepted Answer

The answer of Assume that male and female births are...

Question 17

A recent survey found that 70% of all adults over 50 wear sunglasses for driving. In a random sample of 10adults over 50, what is the probability that at least six wear sunglasses?

Accepted Answer

This problem can be solved using the binomial probability formula, which is P(X = k) = (nCk) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successful trials, p is the probability of success on a single trial, and nCk is the combination of n items taken k at a time. Here, n = 10, p = 0.70, and we're interested in the probability of at least 6 successes (k ≥ 6), which means we need to calculate the sum of probabilities for k = 6, 7, 8, 9, and 10. Instead of calculating each probability individually and summing them up, recognizing that this is a common type of problem, we can use statistical software or tables that have these cumulative probabilities pre-calculated. The result of this cumulative probability for at least 6 successes out of 10 with a success probability of 0.70 is approximately 0.850, indicating that there's an 85% chance that at least 6 out of 10 adults over 50 in a random sample will wear sunglasses for driving.

Question 18

According to government data, the probability that an adult was never in a museum is 20%. In a randomsurvey of 70 adults, what is the mean and standard deviation of the number that were never in a museum?

Accepted Answer

The mean is calculated as the total number of adults (70) multiplied by the probability of never being in a museum (20% or 0.2), which equals 14. The standard deviation is calculated using the formula for the standard deviation of a binomial distribution, $\sqrt{n \cdot p \cdot (1-p)}$, where $n$ is the number of trials (70), and $p$ is the probability of success (0.2). This calculation gives $\sqrt{70 \cdot 0.2 \cdot 0.8} = 3.346640106$.

Question 19

The probability that an individual has 20-20 vision is 0.1. In a class of 10 students, what is the mean andstandard deviation of the number with 20-20 vision in the class?

Accepted Answer

The mean of a binomial distribution is given by $n \times p$, where $n$ is the number of trials (in this case, 10 students) and $p$ is the probability of success (in this case, 0.1 for having 20-20 vision). Thus, the mean is $10 \times 0.1 = 1$. The standard deviation is given by $\sqrt{n \times p \times (1-p)}$, which calculates to $\sqrt{10 \times 0.1 \times 0.9} = 0.9486833$.

Question 20

The probability that a football game will go into overtime is 15%. In 260 randomly selected football games,what is the mean and the standard deviation of the number that went into overtime?

Accepted Answer

The mean is calculated as 260 games * 15% = 39 games. The standard deviation is calculated using the formula for the standard deviation of a binomial distribution, which is sqrt(n*p*(1-p)) = sqrt(260*0.15*0.85) ≈ 5.75760367.

Quiz 6: Discrete Probability Distributions