Question 1

In the instructor's answer book for a mathematics text, 10% of the answers are incorrect. Use the Poisson approximation to express the probability that there are exactly 2 incorrect answers for a homework set with 50 problems.

Accepted Answer

The answer of In the instructor's answer book for a...

Question 2

In a batch of 100 cell phones, there are, on average, 5 defective ones. If a random sample of 30 is selected, find the probability of 2 defective ones.

Accepted Answer

This is a binomial distribution problem with n=30, p=0.05 (probability of a defective phone), and we want to find the probability of 2 defective ones. Using the binomial probability formula, we get:

P(X=2) = (30 choose 2) (0.05)^2 (1-0.05)^28
P(X=2) = 435 (0.0025) (0.9464)
P(X=2) = 0.3163

Therefore, the probability of 2 defective phones is 0.3163 or approximately 0.32. So, the best choice is C.

Question 3

Use the multinomial formula to find the probability for the following situation. X₁ = 2, X₂ = 1, X₃ = 3 when n = 6, p₁ = 0.3, p₂ = 0.5, p₃ = 0.2

Accepted Answer

Using the multinomial formula, the probability of the given situation is: P(X₁ = 2, X₂ = 1, X₃ = 3) = (6 choose 2,1,3) * (0.3^2 * 0.5 * 0.2^3) = 0.16632 Therefore, the answer is A.

Question 4

A bag contains 30 white marbles and 40 black marbles. If 5 marbles are chosen, what is the probability that there will be 2 white marbles and 3 black marbles?

Accepted Answer

The answer of A bag contains 30 white marbles and...

Question 5

A certain type of battery has a 0.5% failure rate. Find the probability that a shipment of 1,000 batteries has more than two defective batteries.

Accepted Answer

The probability of more than two defective batteries can be found using the Poisson approximation to the binomial distribution. With a failure rate of 0.5% and 1,000 batteries, the expected number of failures (λ) is 5. The probability of having 0, 1, or 2 defective batteries is calculated using the Poisson probability formula, and then subtracted from 1 to find the probability of having more than two defective batteries, which is approximately 0.875.

Question 6

The probability that a Poisson random variable X is equal to 4, where  = 10, is

Accepted Answer

The probability mass function (PMF) of a Poisson random variable is given by:

P(X=k) = (lambda^k * e^(-lambda)) / k!

where lambda is the mean and variance of the Poisson distribution.

In this case, lambda = 10 (given).

So,

P(X=4) = (10^4 * e^(-10)) / 4!

= (10000/24) * e^(-10)

= 0.018132788...

Rounding to 4 decimal places gives:

P(X=4) = 0.0181

which is closest to the value in choice C:

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Question 7

On a Saturday evening in Chicago, 34% of the people go out to dinner, 18% go out to see a movie, 13% go out to a party, and 35% stay home. If five people are randomly selected on Monday morning, what is the probability that all but one went out Saturday night?

Accepted Answer

The answer of On a Saturday evening in Chicago, 34%...

Question 8

A survey of 11 people at a bookstore showed that five had purchased a hardcover book. If seven people are selected at random, what is the probability that four of them had purchased a hardcover book?

Accepted Answer

We can use the binomial probability formula:

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

where:
n = sample size = 7
k = number of people who purchased hardcover book = 4
p = probability of purchasing a hardcover book = 5/11

Plugging in values, we get:

P(X = 4) = (7 choose 4) * (5/11)^4 * (6/11)^3

P(X = 4) = (35) * (0.041) * (0.271)

P(X = 4) = 0.303

Therefore, the probability that four of the seven selected people had purchased a hardcover book is 0.303. The answer is choice C.

Question 9

A school club consists of 30 male students and 13 female students. If 3 students are selected to represent the club in the student government, what is the probability 1 will be male and 2 will be female?

Accepted Answer

The answer of A school club consists of 30 male...

Quiz 5: Discrete Probability Distributions