Question 1

From past experience, it is known 90% of one-year-old children can distinguish their mother's voice from the voice of a similar-sounding female. A random sample of 20 one-year-olds are given this voice recognition test.
a. Find the probability that at least three children do not recognize their mother's voice.
b. Find the probability that all 20 children recognize their mother's voice.
c. Let the random variable x denote the number of children who do not recognize their mother's voice. Find the mean of x.
d. Let the random variable x denote the number of children who do not recognize their mother's voice. Find the variance of x.
e. Find the probability that, at most, four children do not recognize their mother's voice.

Accepted Answer

The answer of From past experience, it is known 90%...

Question 2

A warehouse contains six parts made by company A and ten parts made by company B. If four parts are selected at random from the warehouse, the probability that none of the four parts is from company A is approximately 0.1154.

Accepted Answer

Let's use the formula for probability of an event:
P(event) = (number of successful outcomes) / (total number of outcomes)

In this case, the "successful outcomes" are selecting four parts from company B, and the "total number of outcomes" is selecting any four parts from the warehouse. We can use combinations to calculate these numbers:

- Number of ways to choose 4 parts from company B: C(10,4) = 210
- Number of ways to choose any 4 parts from the warehouse: C(16,4) = 1820

So the probability of selecting 4 parts from company B is:
P(BBBB) = 210 / 1820 = 21 / 182

To check the given probability of approximately 0.1154, we can use a calculator or software to find the closest value of P(BBBB) that matches:

- P(BBBB) ≈ 0.115372

This is very close to the given probability, so we can say that the statement is true: the probability of selecting four parts at random from the warehouse and getting none from company A is approximately 0.1154.

Question 3

A fellow student tells you he is working with a discrete random variable, x, which takes on integer values from 0 to 50 and has a mean of 40 and a variance of 10. Is x a binomial random variable?

Accepted Answer

The answer of A fellow student tells you he is...

Question 4

The Poisson probability tables list the probabilities of x occurrences in a Poisson process for various values of  , the mean number of occurrences.

Accepted Answer

The Poisson probability tables are used to calculate the probabilities of x occurrences in a Poisson process for various values of λ, the mean number of occurrences. Therefore, the statement is true.

Question 5

The taste test for PTC (phenylthiourea) is a common class demonstration in the study of genetics. It is known that 70% of Canadians are "tasters" and 30% are "non-tasters." Suppose a genetics class of size 20 does the test to see if they match the Canadian percentage of "tasters" and "non-tasters." (Assume the assignment of students to classes constitutes a random process.)
a. What is the probability distribution of the random variable x, the number of "non-tasters" in the class?
b. Find P(3 < x < 9).
c. Find the mean of x.
d. Find the variance of x.

Accepted Answer

The answer of The taste test for PTC (phenylthiourea) is...

Question 6

A quiz consists of 15 multiple-choice questions. Each question has five choices, with exactly one correct choice. A student, totally unprepared for the quiz, guesses on each of the 15 questions.
a. How many questions should the student expect to answer correctly?
b. What is the standard deviation of the number of questions answered correctly?
c. If at least nine questions must be answered correctly to pass the quiz, what is the chance the student passes?

Accepted Answer

The answer of A quiz consists of 15 multiple-choice questions....

Question 7

Suppose 40% of the TV sets in use in Canada on a particular night were tuned into game 7 of the Stanley Cup Playoffs.
a. If we were to take a sample of six in-use TV sets that night, what is the probability exactly three are tuned to the Stanley Cup Playoffs?
b. If, instead, the sample consisted of 15 in-use TVs, what is the probability 5 or more are tuned to the Stanley Cup Playoffs?

Accepted Answer

The answer of Suppose 40% of the TV sets in...

Question 8

The Poisson distribution is appropriate to determine the probability of a given number of defective items in a shipment.

Accepted Answer

The Poisson distribution is used for modeling the number of events occurring within a fixed interval of time or space, typically when these events are rare. For defective items in a shipment, the binomial distribution is more appropriate as it deals with a fixed number of trials and two possible outcomes (defective or not defective).

Question 9

The Poisson parameter  is the mean number of occurrences of an event per unit of time or space during the Poisson process.

Accepted Answer

The Poisson parameter is indeed the mean number of occurrences of an event per unit of time or space during the Poisson process.

Question 10

Hypergeometric probability distributions are examples of discrete probability distributions.

Accepted Answer

Hypergeometric probability distributions describe the probability of the number of successes in a fixed sample size drawn without replacement from a finite population of different types of objects. As the possible outcomes are finite and countable, hypergeometric distributions are a type of discrete probability distribution.

Question 11

A hypergeometric probability distribution shows the probabilities associated with possible values of a discrete random variable when these values are generated by sampling with replacement, and the probability of success, therefore, changes from one trial to the next.

Accepted Answer

A hypergeometric probability distribution shows the probabilities associated with possible values of a discrete random variable when these values are generated by sampling without replacement, and the probability of success is fixed for all trials.

Question 12

A warehouse contains six parts made by company A and ten parts made by company B. If four parts are selected at random from the warehouse, the probability that all four parts are from company B is approximately 0.008.

Accepted Answer

The answer of A warehouse contains six parts made by...

Question 13

A Poisson process is the occurrence of a series of events of a given type in a random pattern over time or space such that (1) the number of occurrences within a specified time or space can equal any integer between zero and infinity, (2) the number of occurrences within one unit of time or space is independent of that in any other such (nonoverlapping) unit, and (3) the probability of occurrences is the same in all such units.

Accepted Answer

The description provided matches the definition of a Poisson process.

Question 14

The mean of a Poisson distribution variable, where  is the average number of successes occurring in a specified interval, is  .

Accepted Answer

The answer of The mean of a Poisson distribution variable,...

Question 15

An oil firm plans to drill 20 wells, each having a probability 0.2 of striking oil. Each well costs $20,000 to drill; a well that strikes oil will bring in $750,000 in revenue. Find the expected gain from the 20 wells.

Accepted Answer

The answer of An oil firm plans to drill 20...

Question 16

The Poisson random variable is the number of successes achieved when a random sample of size n is drawn without replacement from a population of size N within which M units have the characteristic that denotes success.

Accepted Answer

The Poisson random variable is the number of occurrences of an event within a fixed interval of time or space, and is not dependent on a sample size or population size. It is characterized by a single parameter λ, which represents the average rate of events occurring in the interval. The Poisson distribution is often used to model rare events, such as the number of accidents in a day or the number of defects in a manufacturing process.

Question 17

The hypergeometric probability distribution formula calculates the probability of x successes when a random sample of size n is drawn without replacement from a population of size N within which M units have the characteristic that denotes success, and N - M units have the characteristic that denotes failure.

Accepted Answer

This is a correct description of the hypergeometric probability distribution formula.

Question 18

Hypergeometric probability distributions are examples of continuous probability distributions.

Accepted Answer

Hypergeometric probability distributions are examples of discrete probability distributions, not continuous probability distributions.

Question 19

A warehouse contains six parts made by company A and ten parts made by company B. If four parts are selected at random from the warehouse, the probability that one of the four parts is from company B is approximately 0.1099.

Accepted Answer

This can be solved using the binomial probability distribution with n = 4 (total number of parts selected) and p = 10/16 (probability of selecting a part from company B). The probability of selecting exactly one part from company B is approximately 0.1099, so the answer is true.

Question 20

The mean and variance of a Poisson distribution variable are equal.

Accepted Answer

This statement is true. The mean and variance of a Poisson distribution variable are both equal to λ, the parameter of the Poisson distribution.

Quiz 5: Several Useful Discrete Distributions