Question 1

The probability that an individual is left-handed is 0.12. In a class of 39 students, what is the probability of finding five left-handers?

Accepted Answer

This is a binomial probability problem with n=39 and p=0.12. We want to find the probability of getting exactly 5 left-handers. This can be calculated using the binomial probability formula:

P(X=5) = (39 choose 5) * 0.12^5 * (1-0.12)^(39-5)

Using a calculator, we get P(X=5) = 0.186. Therefore, the probability of finding five left-handers is 0.186. Choice A is the correct answer.

Question 2

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

Accepted Answer

The probability of at least six adults wearing glasses can be calculated using the binomial probability formula or cumulative binomial probability tables. Since calculating each probability for exactly 6, 7, 8, 9, and 10 adults wearing glasses and summing them up is cumbersome, using a binomial probability calculator or tables is more efficient. The result is approximately 0.850, indicating a high likelihood that at least six out of ten adults over 50 in a random sample will wear glasses for driving.

Question 3

In a recent survey, 64% of the community favored building a police substation in their neighborhood. If 14 citizens are chosen, find the probability that exactly 8 of them favor the building of the police substation.

Accepted Answer

This is a binomial probability problem where n = 14 (number of trials), k = 8 (number of successes), and p = 0.64 (probability of success). The formula for binomial probability is P(X = k) = (nCk) * p^k * (1-p)^(n-k). Plugging in the values: P(X = 8) = (14C8) * (0.64)^8 * (0.36)^6 ≈ 0.184.

Question 4

Assume that male and female births are equally likely and that the birth of any child does not affect the
probability of the gender of any other children. Suppose that 500 couples each have a baby; find the mean and
standard deviation for the number of girls in the 500 babies.

Accepted Answer

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

Question 5

According to government data, the probability that a woman between the ages of 25 and 29 was never married is 40%. In a random survey of 10 women in this age group, what is the mean and standard deviation of the
Number that never married?

Accepted Answer

The mean of the number that never married is calculated as the product of the total number of women surveyed and the probability of a woman never being married, which is $10 \times 0.40 = 4$. The standard deviation is calculated using the formula for the standard deviation of a binomial distribution, $\sqrt{n \times p \times (1-p)}$, which gives $\sqrt{10 \times 0.40 \times (1-0.40)} = \sqrt{10 \times 0.40 \times 0.60} = \sqrt{2.4} = 1.55$.

Question 6

According to government data, the probability that a woman between the ages of 25 and 29 was never married is 40%. In a random survey of 10 women in this age group, what is the probability that two or fewer were never
Married?

Accepted Answer

The probability that two or fewer were never married is the sum of the probabilities that 0, 1, or 2 were never married. The probability that 0 were never married is (0.6)^10 = 0.000977. The probability that 1 was never married is 10 * (0.4)^1 * (0.6)^9 = 0.01296. The probability that 2 were never married is 45 * (0.4)^2 * (0.6)^8 = 0.07558. The sum of these probabilities is 0.000977 + 0.01296 + 0.07558 = 0.089517, which is approximately 0.09.

Question 7

The probability that a tennis set will go to a tiebreaker is 16%. In 220 randomly selected tennis sets, what is the mean and the standard deviation of the number of tiebreakers?

Accepted Answer

The mean of the number of tiebreakers is found by multiplying the probability of a tiebreaker by the number of sets: $220 \times 0.16 = 35.2$. 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 number of trials and $p$ is the probability of success: $\sqrt{220 \times 0.16 \times (1-0.16)} = \sqrt{35.2 \times 0.84} \approx 5.44$.

Question 8

The probability that a tennis set will go to a tie-breaker is 14%. What is the probability that two of three sets will go to tie-breakers?

Accepted Answer

The probability of two sets going to tie-breakers is calculated using the binomial probability formula: P(X=k) = C(n,k) * p^k * (1-p)^(n-k), where n=3, k=2, and p=0.14. This results in 3 * (0.14)^2 * (0.86)^1 = 0.051.

Question 9

Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Find the probability of exactly nine boys in ten births.

Accepted Answer

The probability of exactly nine boys in ten births can be calculated using the binomial probability formula, which is $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$, where $n$ is the number of trials (in this case, 10 births), $k$ is the number of successful outcomes desired (in this case, 9 boys), and $p$ is the probability of success on any given trial (in this case, 0.5, since each birth is equally likely to be a boy or a girl). Plugging in the values, we get $P(X=9) = \binom{10}{9}(0.5)^9(0.5)^{10-9} = 10(0.5)^{10} = 0.010$.

Question 10

According to police sources, a car with a certain protection system will be recovered 90% of the time. If 600 stolen cars are randomly selected, what is the mean and standard deviation of the number of cars recovered
After being stolen?

Accepted Answer

The mean of the number of cars recovered is calculated as 90% of 600, which is 0.90 * 600 = 540. 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 (600), p is the probability of success (0.90), and (1-p) is the probability of failure (0.10). Thus, the standard deviation is sqrt(600*0.90*0.10) = sqrt(54) = 7.35.

Question 11

Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Find the probability of at most three boys in ten births.

Accepted Answer

We can use the binomial distribution to solve this problem. Let X be the number of boys in ten births. Then X follows a binomial distribution with parameters n=10 and p=0.5. The probability of at most three boys in ten births is:

P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

Using the binomial probability formula, we get:

P(X = k) = (10 choose k) * 0.5^k * 0.5^(10-k)

Thus,

P(X = 0) = (10 choose 0) * 0.5^0 * 0.5^10 = 0.00098
P(X = 1) = (10 choose 1) * 0.5^1 * 0.5^9 = 0.00977
P(X = 2) = (10 choose 2) * 0.5^2 * 0.5^8 = 0.04395
P(X = 3) = (10 choose 3) * 0.5^3 * 0.5^7 = 0.11719

Therefore,

P(X ≤ 3) = 0.00098 + 0.00977 + 0.04395 + 0.11719 = 0.172

So the answer is A, 0.172. The number of ways to have at most three boys in ten births is the sum of the number of ways to have 0, 1, 2, or 3 boys in ten births.

The number of ways to have 0 boys in ten births is $\binom{10}{0} = 1$. The number of ways to have 1 boy in ten births is $\binom{10}{1} = 10$. The number of ways to have 2 boys in ten births is $\binom{10}{2} = 45$. The number of ways to have 3 boys in ten births is $\binom{10}{3} = 120$.

The total number of ways to have any combination of boys and girls in ten births is $2^{10} = 1024$.

So the probability of at most three boys in ten births is $$\frac{1 + 10 + 45 + 120}{1024} = \frac{196}{1024} \approx \boxed{0.172}$$

Question 12

According to police sources, a car with a certain protection system will be recovered 89% of the time. Find the probability that 3 of 7 stolen cars will be recovered.

Accepted Answer

The answer of According to police sources, a car with...

Question 13

Fifty percent of the people that get mail-order catalogs order something. Find the probability that exactly two of 10 people getting these catalogs will order something.

Accepted Answer

This is a binomial probability problem where n = 10 (number of trials), k = 2 (number of successes), and p = 0.5 (probability of success on a single trial). The formula for binomial probability is P(X = k) = (nCk) * p^k * (1-p)^(n-k), where nCk is the combination of n items taken k at a time. Plugging in the values: P(X = 2) = (10C2) * (0.5)^2 * (0.5)^(10-2) = 45 * 0.25 * 0.00390625 = 0.044.

Question 14

The probability that a house in an urban area will be burglarized is 5%. If 82 houses are randomly selected, what is the probability that none of the houses will be burglarized?

Accepted Answer

The probability that a house will not be burglarized is 1 - 0.05 = 0.95. For 82 houses, the probability that none will be burglarized is 0.95^82 ≈ 0.015.

Question 15

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

Accepted Answer

The probability of passing the test (answering at least 8 out of 10 questions correctly) by guessing can be calculated using the binomial probability formula. The probability of success (getting a question right) is 0.5 for each question since it's a true or false question. The probabilities of getting exactly 8, 9, or 10 questions right need to be calculated and then summed up. The calculations yield a probability of approximately 0.055.

Question 16

A recent survey found that 63% of all adults over 50 wear glasses for driving. In a random sample of 100 adults over 50, what is the mean and standard deviation of those that wear glasses?

Accepted Answer

The mean of those that wear glasses is found by multiplying the percentage (63%) by the sample size (100), giving 63. 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 sample size (100) and $p$ is the probability (0.63), resulting in $\sqrt{100 \cdot 0.63 \cdot (1-0.63)} = \sqrt{23.31} \approx 4.83$.

Question 17

A test consists of 100 multiple choice questions, each with five possible answers, only one of which is correct. Find the mean and the standard deviation of the number of correct answers.

Accepted Answer

The answer of A test consists of 100 multiple choice...

Question 18

A test consists of 10 multiple choice questions, each with five possible answers, one of which is correct. To pass the test a student must get 60% or better on the test. If a student randomly guesses, what is the probability that
The student will pass the test?

Accepted Answer

To pass, the student needs at least 6 correct answers. The probability of guessing one question correctly is 1/5. Using the binomial probability formula, calculate the probability of getting exactly 6, 7, 8, 9, and 10 correct answers and sum them. The result is approximately 0.006.

Question 19

The probability that an individual is left-handed is 0.16. In a class of 10 students, what is the mean and standard deviation of the number of left-handers in the class?

Accepted Answer

The mean of left-handers in the class is calculated as the total number of students multiplied by the probability of being left-handed, which is $10 \times 0.16 = 1.6$. The standard deviation is calculated using the formula for the standard deviation of a binomial distribution, $\sqrt{n \times p \times (1-p)}$, which gives $\sqrt{10 \times 0.16 \times (1-0.16)} = 1.16$.

Question 20

According to government data, the probability that a woman between the ages of 25 and 29 was never married is 40%. In a random survey of 10 women in this age group, what is the probability that at least eight were
Married?

Accepted Answer

The probability of at least eight women being married is the complement of the probability of seven or fewer women being married. The probability of seven or fewer women being married is the sum of the probabilities of zero, one, two, three, four, five, six, or seven women being married. Using the binomial distribution, we can calculate these probabilities as follows:P(X = 0) = (0.6)^10 = 0.000006P(X = 1) = 10 * (0.6)^9 * (0.4)^1 = 0.000096P(X = 2) = 45 * (0.6)^8 * (0.4)^2 = 0.000648P(X = 3) = 120 * (0.6)^7 * (0.4)^3 = 0.00288P(X = 4) = 210 * (0.6)^6 * (0.4)^4 = 0.009072P(X = 5) = 252 * (0.6)^5 * (0.4)^5 = 0.02304P(X = 6) = 210 * (0.6)^4 * (0.4)^6 = 0.0432P(X = 7) = 120 * (0.6)^3 * (0.4)^7 = 0.0648Therefore, the probability of at least eight women being married is:1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)) = 1 - 0.167 = 0.833Therefore, the correct answer is A.

Quiz 4: Discrete Probability Distributions