Question 1

One hundred policyholders file claims with CareFree Insurance. Ten of these claims are fraudulent. Claims manager Earl Evans randomly selects four of the one hundred claims for thorough investigation. If x represents the number of fraudulent claims in Earl's sample, x has a ___ distribution.
A) continuous
B) normal
C) binomial
D) hypergeometric
E) exponential

Accepted Answer

The situation describes a hypergeometric distribution because it involves selecting a sample without replacement from a finite population that contains exactly two types of objects (fraudulent and non-fraudulent claims), which fits the definition of a hypergeometric distribution.

Question 2

The number of defects per 1,000 feet of extruded plastic pipe is best modelled with the ___.
A) Poisson distribution
B) Pascal distribution
C) binomial distribution
D) hypergeometric distribution
E) geometric distribution

Accepted Answer

The Poisson distribution is appropriate for modeling the number of rare events occurring in a fixed interval. In this case, defects are expected to be rare relative to the length of the piping, and the Poisson distribution is a good choice for modeling the number of defects.

Question 3

The Human Resource manager of a corporation documents the frequency with which employees call in sick. The Human Resource manager develops the following probability distribution for the number of time staff called in sick in a particular month:
  Determine the expected value and the standard deviation of the probability distribution.

Accepted Answer

The answer of The Human Resource manager of a corporation...

Question 4

The probability that a student will pass an exam on any trial is 0.7. What is the standard deviation of the number of trials needed to pass the exam?
A) 0.540
B) 0.548
C) 0.782
D) 0.880
E) 0.002

Accepted Answer

The answer of The probability that a student will pass...

Question 5

A basketball player makes 80% of her free throws. Suppose this probability is the same for each free throw she attempts. What is probability that she makes more than 5 free throw attempts this season?
A) 0.32
B) 0.032
C) 0.0032
D) 0.00032
E) 0.000030

Accepted Answer

The answer of A basketball player makes 80% of her...

Question 6

A basketball player makes 80% of her free throws. Suppose this probability is the same for each free throw she attempts. The probability that she doesn't make a free throw until her fifth attempt this season is
A) 0.32768.
B) 0.08192.
C) 0.00128.
D) 0.00032.
E) 0.00004.

Accepted Answer

The probability that she doesn't make a free throw until her fifth attempt is calculated using the geometric distribution: (0.2)^4 * 0.8 = 0.00128.

Question 7

Which of the following random variables is geometric?
A) the number of phone calls received in a one-hour period
B) the number of cards I need to deal from a deck of 52 cards that has been thoroughly shuffled so that at least one of the cards is a heart
C) the number of digits I will read beginning at a randomly selected starting point in a table of random digits until I find a 7
D) all of the above
E) none of the above

Accepted Answer

A geometric random variable represents the number of trials needed to achieve the first success, where the probability of success remains constant across trials. Option C fits this definition as we are counting the number of digits we need to read until we find a 7 which is the first "success". Options A and B do not fit the definition of a geometric random variable. Option A represents a Poisson distribution, and option B represents a hypergeometric distribution.

Question 8

One hundred policyholders file claims with CareFree Insurance. Ten of these claims are fraudulent. Claims manager Earl Evans randomly selects four of the one hundred claims for thorough investigation. If x represents the number of fraudulent claims in Earl's sample, x has a ___.
A) normal distribution
B) hypergeometric distribution, but may be approximated by a binomial
C) binomial distribution, but may be approximated by a normal
D) binomial distribution, but may be approximated by a Poisson
E) exponential distribution

Accepted Answer

The situation described is an example of a hypergeometric distribution, as we are sampling without replacement from a finite population of two classes (fraudulent and legitimate claims). However, for a large population and a small sample size (which is the case here), the hypergeometric distribution can be approximated by a binomial distribution. Therefore, we choose option B.

Question 9

Ten policyholders file claims with CareFree Insurance. Three of these claims are fraudulent. Claims manager Earl Evans randomly selects three of the ten claims for thorough investigation. If x represents the number of fraudulent claims in Earl's sample, P(x=1) is ___.
A) 0.5250
B) 0.4410
C) 0.3000
D) 0.6957
E) 0.9957

Accepted Answer

The probability of selecting exactly 1 fraudulent claim out of 3 chosen from 10, where 3 are fraudulent and 7 are legitimate, can be calculated using combinations. The probability is found by dividing the number of ways to choose 1 fraudulent and 2 legitimate claims by the total ways to choose any 3 claims from 10. This is calculated as: $\frac{{\binom{3}{1} \times \binom{7}{2}}}{{\binom{10}{3}}} = \frac{{3 \times 21}}{{120}} = 0.525$.

Question 10

On Monday mornings, customers arrive at the coffee shop drive thru at the rate of 6 cars per fifteen-minute interval. Using the Poisson distribution, the probability that five cars will arrive during the next fifteen-minute interval is ___.
A) 0.1008
B) 0.0361
C) 0.1339
D) 0.1606
E) 0.5000

Accepted Answer

The answer of On Monday mornings, customers arrive at the...

Question 11

In a certain communications system, there is an average of 1 transmission error per 10 seconds. Assume that the distribution of transmission errors is Poisson. The probability of 1 error in a period of one-half minute is approximately ___
A) 0.1493
B) 0.3333
C) 0.3678
D) 0.1336
E) 0.03

Accepted Answer

The answer of In a certain communications system, there is...

Question 12

Suppose X has a geometric distribution with probability 0.3 of success on each observation. The probability that X is 4 is
A) 0.0081.
B) 0.0189.
C) 0.1029.
D) 0.2401.
E) 0.2409.

Accepted Answer

The answer of Suppose X has a geometric distribution with...

Question 13

The number of bags arriving on the baggage claim conveyor belt in a 3-minute time period would best be modelled with the ___.
A) binomial distribution
B) hypergeometric distribution
C) Poisson distribution
D) hyperbinomial distribution
E) geometric distribution

Accepted Answer

The Poisson distribution is used to model the number of events that occur in a fixed interval of time or space, given the average rate of occurrence per interval. In this case, we are interested in the number of bags arriving on the conveyor belt in a fixed 3-minute time period, and the arrival of each bag is a random event that occurs independently of other events. The Poisson distribution is appropriate for this type of scenario. The other choices are not appropriate because they either do not apply to the scenario (geometric and hyperbinomial distributions) or require additional information that is not given (binomial and hypergeometric distributions).

Question 14

The probability that a student will pass an exam on any trial is 0.7. What is the expected number of trials needed to pass the exam?
A) 0.420
B) 0.428
C) 0.429
D) 1.420
E) 1.430

Accepted Answer

The expected value of a geometric distribution (number of trials until success) with probability p is 1/p.
Therefore, in this case, the expected number of trials until passing the exam is 1/0.7 = 1.429, which is closest to option C (0.429) after rounding to three decimal places.

Question 15

If sampling is performed without replacement, the hypergeometric distribution should be used. However, the binomial may be used to approximate this if ___.
A) n ≥ 5%N
B) n < 5%N
C) the population size is very small
D) there are more than two possible outcomes of each trial
E) the outcomes are continuous

Accepted Answer

When sampling is performed without replacement, the hypergeometric distribution should be used. However, the binomial distribution can be used to approximate the hypergeometric distribution when the sample size is small relative to the population size (n < 5%N).

Question 16

The probability that a student will pass an exam on any trial is 0.7. What is the probability that he will pass on the first trial?
A) 0.7
B) 0.21
C) 0.06
D) 0.021
E) 1.00

Accepted Answer

The probability of passing on the first trial is the same as the overall probability of passing, which is 0.7. Therefore, the answer is A.

Question 17

The number of cars arriving at a toll booth in five-minute intervals is Poisson distributed with a mean of 3 cars arriving in five-minute time intervals. The probability of 5 cars arriving over a five-minute interval is ___.
A) 0.0940
B) 0.0417
C) 0.1500
D) 0.1008
E) 0.2890

Accepted Answer

The answer of The number of cars arriving at a...

Question 18

The number of cars arriving at a toll booth in five-minute intervals is Poisson distributed with a mean of 3 cars arriving in five-minute time intervals. The probability of 3 cars arriving over a five-minute interval is ___.
A) 0.2700
B) 0.0498
C) 0.2240
D) 0.0001
E) 0.0020

Accepted Answer

The answer of The number of cars arriving at a...

Question 19

Assume that a random variable has a Poisson distribution with a mean of 5 occurrences per ten minutes. The number of occurrences per hour follows a Poisson distribution with λ equal to ___.
A) 5
B) 60
C) 30
D) 10
E) 20

Accepted Answer

We know that the mean number of occurrences in 10 minutes is 5. Therefore, the mean number of occurrences in 60 minutes (1 hour) would be (5 occurrences/10 minutes) x 6 = 30 occurrences. Therefore, the number of occurrences per hour follows a Poisson distribution with λ = 30.

Question 20

On Monday mornings, customers arrive at the coffee shop drive thru at the rate of 6 cars per fifteen-minute interval. Using the Poisson distribution, the probability that five cars will arrive during the next five-minute interval is ___.
A) 0.1008
B) 0.0361
C) 0.1339
D) 0.1606
E) 0.3610

Accepted Answer

Since the rate of arrival is given for every 15 minutes, we need to adjust it to the 5-minute interval. 
Let λ be the average number of arrivals in a 5-minute interval. 
Then, λ = (6/15) * 5 = 2.
Using the Poisson distribution: 
P(X = 5) = (e^(-λ) * λ^5) / 5! = (e^(-2) * 2^5) / 120 = 0.0361. 
Therefore, the probability that five cars will arrive during the next five-minute interval is 0.0361 or approximately 3.61%.

Quiz 7: Discrete Probability Distributions

One hundred policyholders file claims with CareFree Insurance. Ten of these claims are fraudulent. Claims manager Earl Evans randomly selects four of the one hundred claims for thorough investigation. If x represents the number of fraudulent claims in Earl's sample, x has a ___ distribution.

The number of defects per 1,000 feet of extruded plastic pipe is best modelled with the ___.

The probability that a student will pass an exam on any trial is 0.7. What is the standard deviation of the number of trials needed to pass the exam?

A basketball player makes 80% of her free throws. Suppose this probability is the same for each free throw she attempts. What is probability that she makes more than 5 free throw attempts this season?

A basketball player makes 80% of her free throws. Suppose this probability is the same for each free throw she attempts. The probability that she doesn't make a free throw until her fifth attempt this season is

Which of the following random variables is geometric?

One hundred policyholders file claims with CareFree Insurance. Ten of these claims are fraudulent. Claims manager Earl Evans randomly selects four of the one hundred claims for thorough investigation. If x represents the number of fraudulent claims in Earl's sample, x has a ___.

Ten policyholders file claims with CareFree Insurance. Three of these claims are fraudulent. Claims manager Earl Evans randomly selects three of the ten claims for thorough investigation. If x represents the number of fraudulent claims in Earl's sample, P(x=1) is ___.

On Monday mornings, customers arrive at the coffee shop drive thru at the rate of 6 cars per fifteen-minute interval. Using the Poisson distribution, the probability that five cars will arrive during the next fifteen-minute interval is ___.

In a certain communications system, there is an average of 1 transmission error per 10 seconds. Assume that the distribution of transmission errors is Poisson. The probability of 1 error in a period of one-half minute is approximately ___

Suppose X has a geometric distribution with probability 0.3 of success on each observation. The probability that X is 4 is

The number of bags arriving on the baggage claim conveyor belt in a 3-minute time period would best be modelled with the ___.

The probability that a student will pass an exam on any trial is 0.7. What is the expected number of trials needed to pass the exam?

If sampling is performed without replacement, the hypergeometric distribution should be used. However, the binomial may be used to approximate this if ___.

The probability that a student will pass an exam on any trial is 0.7. What is the probability that he will pass on the first trial?

The number of cars arriving at a toll booth in five-minute intervals is Poisson distributed with a mean of 3 cars arriving in five-minute time intervals. The probability of 5 cars arriving over a five-minute interval is ___.

The number of cars arriving at a toll booth in five-minute intervals is Poisson distributed with a mean of 3 cars arriving in five-minute time intervals. The probability of 3 cars arriving over a five-minute interval is ___.

Assume that a random variable has a Poisson distribution with a mean of 5 occurrences per ten minutes. The number of occurrences per hour follows a Poisson distribution with λ equal to ___.

On Monday mornings, customers arrive at the coffee shop drive thru at the rate of 6 cars per fifteen-minute interval. Using the Poisson distribution, the probability that five cars will arrive during the next five-minute interval is ___.