Question 1

If there are 20 typographical errors randomly distributed in a 250-page document, find the probability that a given page contains exactly two errors.

Accepted Answer

The answer of If there are 20 typographical errors randomly...

Question 2

Use the multinomial formula to find the probability for the situation in which n = 5, X₁ = 2, X₂ = 2, X₃ = 1, and, p₁ = 0.4, p₂ = 0.4, p₃ = 0.2.

Accepted Answer

The multinomial formula is given by: P(X1 = x1, X2 = x2, ..., Xk = xk) = n! / (x1! x2! ... xk!) * p1^x1 * p2^x2 * ... * pk^xk where n is the total number of trials, k is the number of possible outcomes, xi is the number of trials that result in outcome i, and pi is the probability of outcome i. In this case, we have k = 3 (since there are three possible outcomes: ₁, ₂, and ₃), n = 5, x1 = 2, x2 = 2, and x3 = 1. We are also given the probabilities p1 = 0.4, p2 = 0.4, and p3 = 0.2. Substituting these values into the multinomial formula, we get: P(X₁ = 2, X₂ = 2, X₃ = 1) = 5! / (2! 2! 1!) * 0.4^2 * 0.4^2 * 0.2^1 = 0.1536 Therefore, the answer is choice C.

Question 3

In a binomial experiment, the outcome of each trial will depend on the ones before it.

Accepted Answer

The outcome of each trial in a binomial experiment is independent of the ones before it, meaning the outcome of one trial does not impact the probability of success or failure in the next trial.

Question 4

In a survey, 60% of the voters support a particular referendum. If 40 voters are chosen at random, find the standard deviation of the number of voters who support the referendum.

Accepted Answer

The answer of In a survey, 60% of the voters...

Question 5

Use the multinomial formula and find the probability for the following data.

Accepted Answer

Since the probabilities provided add up to 1, we use the multinomial formula to find the probability:

P = (11 choose 2,2,2,2,3) * (0.1)^2 * (0.3)^2 * (0.2)^2 * (0.15)^2 * (0.25)^3 = 0.057

Therefore, the correct answer is A) 0.057.

Question 6

Find the probability  : P(5, 6)

Accepted Answer

Since P(n,r) = n!/(n-r)!n^r, P(5,6) = 0 as n=5 and r=6 cannot be accepted. Therefore, the correct answer is C (which suggests none of the above).

Question 7

A computer store has 75 printers of which 25 are laser printers and 50 are ink jet printers. If a group of 10 printers is chosen at random from the store, find the mean and variance of the number of ink jet printers.

Accepted Answer

The mean number of ink jet printers in a group of 10 is calculated using the formula for the mean of a hypergeometric distribution: (n * K) / N = (10 * 50) / 75 = 6.7. The variance is calculated using the formula: [n * (K/N) * (1 - K/N) * (N-n)/(N-1)] = [10 * (50/75) * (25/75) * (65/74)] = 2.2.

Question 8

In a survey, 55% of the voters support a particular referendum. If 30 voters are chosen at random, find the mean and variance for the number of voters who support the referendum.

Accepted Answer

The answer of In a survey, 55% of the voters...

Question 9

The probability that a person will have 0, 1, or 2 dental checkups per year is 0.3, 0.6, and 0.1, respectively. If seven people are picked at random, what is the probability that two will have no checkups, four will have one checkup, and one will have two checkups in the next year?

Accepted Answer

The answer of The probability that a person will have...

Question 10

The probability that federal income tax returns will have 0, 1, or 2 errors is 0.73, 0.23, and 0.04, respectively. If 10 randomly selected returns are audited, what is the probability that eight will have no errors, two will have one error, and none will have two errors?

Accepted Answer

The answer of The probability that federal income tax returns...

Question 11

A binomial distribution is the probabilities of the possible outcomes of a binomial experiment.

Accepted Answer

This is true. A binomial distribution is used to model the probabilities of the possible outcomes of a binomial experiment, which has two possible outcomes (success or failure) and is conducted multiple times under the same conditions.

Question 12

A university has 10,000 students of which 45% are male and 55% are female. If a class of 30 students is chosen at random from the university population, find the mean and variance of the number of male students.

Accepted Answer

The answer of A university has 10,000 students of which...

Question 13

Use the multinomial formula and find the probability for the following data.

Accepted Answer

We don't have enough context or data to apply the multinomial formula. There is no information about the events or the probabilities associated with them. Therefore, we cannot calculate a probability and none of the choices are valid. It is not possible to select the best choice among the given options.

Question 14

In binomial experiments, each trial can be viewed as having only two outcomes.

Accepted Answer

Binomial experiments have only two possible outcomes for each trial, typically designated as success or failure.

Question 15

The failure rate for taking the bar exam in Philadelphia is 41%. If 375 people take the bar exam, what is the mean for the number of failures?

Accepted Answer

The answer of The failure rate for taking the bar...

Question 16

The binomial distribution equations will not give exact probabilities for experiments in which sampling is done without replacement. In those cases, the trials are not independent.

Accepted Answer

The binomial distribution assumes independent trials, which is not the case when sampling is done without replacement. Therefore, the binomial distribution cannot give exact probabilities in such cases.

Question 17

A die is rolled 360 times. Find the standard deviation of the number of times a 3 will be rolled.

Accepted Answer

The probability of rolling a 3 on one throw of the die is 1/6. The expected number of times a 3 will be rolled in 360 throws is 
E(x) = np = 360 * (1/6) = 60
The variance is 
var(x) = npq = 360 * (1/6) * (5/6) = 50
The standard deviation is the square root of the variance, which is approximately 7.1. Therefore, the correct choice is C.

Question 18

A school is sending 17 children to a camp. If 20% of the children in the school are first graders, and the 17 children are selected at random from among all 6 grades at the school, find the mean and variance of the number of first graders chosen?

Accepted Answer

The answer of A school is sending 17 children to...

Question 19

The probability of a success remains the same for each trial in a binomial experiment.

Accepted Answer

The probability of a success in a binomial experiment is constant for each trial. This is one of the defining characteristics of a binomial experiment.

Question 20

A certain large manufacturing facility produces 20,000 parts each week. The manager of the facility estimates that about 1% of the parts they make are defective. What is the variance for the number of defective parts made each week?

Accepted Answer

The answer of A certain large manufacturing facility produces 20,000...

Quiz 5: Discrete Probability Distributions

If there are 20 typographical errors randomly distributed in a 250-page document, find the probability that a given page contains exactly two errors.

Use the multinomial formula to find the probability for the situation in which n = 5, X1 = 2, X2 = 2, X3 = 1, and, p1 = 0.4, p2 = 0.4, p3 = 0.2.

In a binomial experiment, the outcome of each trial will depend on the ones before it.

In a survey, 60% of the voters support a particular referendum. If 40 voters are chosen at random, find the standard deviation of the number of voters who support the referendum.

Use the multinomial formula and find the probability for the following data.

Find the probability : P(5, 6)

A computer store has 75 printers of which 25 are laser printers and 50 are ink jet printers. If a group of 10 printers is chosen at random from the store, find the mean and variance of the number of ink jet printers.

In a survey, 55% of the voters support a particular referendum. If 30 voters are chosen at random, find the mean and variance for the number of voters who support the referendum.

The probability that a person will have 0, 1, or 2 dental checkups per year is 0.3, 0.6, and 0.1, respectively. If seven people are picked at random, what is the probability that two will have no checkups, four will have one checkup, and one will have two checkups in the next year?

The probability that federal income tax returns will have 0, 1, or 2 errors is 0.73, 0.23, and 0.04, respectively. If 10 randomly selected returns are audited, what is the probability that eight will have no errors, two will have one error, and none will have two errors?

A binomial distribution is the probabilities of the possible outcomes of a binomial experiment.

A university has 10,000 students of which 45% are male and 55% are female. If a class of 30 students is chosen at random from the university population, find the mean and variance of the number of male students.

Use the multinomial formula and find the probability for the following data.

In binomial experiments, each trial can be viewed as having only two outcomes.

The failure rate for taking the bar exam in Philadelphia is 41%. If 375 people take the bar exam, what is the mean for the number of failures?

The binomial distribution equations will not give exact probabilities for experiments in which sampling is done without replacement. In those cases, the trials are not independent.

A die is rolled 360 times. Find the standard deviation of the number of times a 3 will be rolled.

A school is sending 17 children to a camp. If 20% of the children in the school are first graders, and the 17 children are selected at random from among all 6 grades at the school, find the mean and variance of the number of first graders chosen?

The probability of a success remains the same for each trial in a binomial experiment.

A certain large manufacturing facility produces 20,000 parts each week. The manager of the facility estimates that about 1% of the parts they make are defective. What is the variance for the number of defective parts made each week?

Use the multinomial formula to find the probability for the situation in which n = 5, X₁ = 2, X₂ = 2, X₃ = 1, and, p₁ = 0.4, p₂ = 0.4, p₃ = 0.2.