Chapter 4

A '_______' random variable assumes a countable number of distinct values such as x1, x2, x3, and so on

'Discrete' random variable

Many experiments fit the conditions of a Bernoulli process.Which of the following fit the conditions of a Bernoulli process? Choose all that apply!

- A drug is either effective or ineffective - A customer defaults or does not default on a loan - A college graduate applies or does not apply to graduate school

For z =.11, what is the corresponding probability?

.5438 Reason: Look at the z =.1 on left hand side and then go over two columns to z =.01, so the corresponding probability is .5438

An economist predicts a 70% chance that country A will perform poorly and a 35% chance that country B will perform poorly. There is also a 20% chance that both countries will perform poorly. What is the probability that country A performs poorly given that country B performs poorly?

.20/.35 =.57

Tiffany Ham's business is thriving in Houston, TX. To reward her team, Tiffany is implementing a performance incentive program. Annual Bonuses begin at $5,000 for excellent performance, $3,000 for good performance, and $1,500 for fair performance, and $0 for poor performance. The probability levels are 0.15, 0.25, 0.40, and 0.05, respectively. What is the expected value of the annual bonus amount for an employee?

$2,100

The z value associated with a probability of .5040 is '____'

0.01

Marketing analysis determined 62% of females between the ages of 25 and 34 years search for green technology and practice being green, as compared to 35% of men in the same age group. What is the probability that a randomly selected woman between the age of 25 and 34 does not search for green technology?

38% probability P(A) = 0.62, so P(Ac) = 1 − P(A) = 1 − 0.62 = 0.38 using the complement rule.

Which theorem can the posterior probability be found using the prior probability and conditional probability?

Bayes' Bayes' Theorem says the posterior probability P(B|A) can be found using the information on the prior probability P(B), along with the conditional probabilities P(A|B) and P(A|Bc).

The standard normal distribution is a special case of the normal distribution with a mean equal to '_________'.

zero

In many instances, we calculate probabilities by referencing data based on the observed outcomes of an experiment. Which probability category is defined as the observed relative frequency with which an event occurs?

Empirical probability

True or false: The joint probability of events A and B is derived as P(A ∩ B) = P(A ∣ B)P(A).

False Reason: The joint probability of events A and B is derived as P(A ∩ B) = P(A ∣ B)P(B).

What are some commonly used terms for the normal distribution?

Gaussian distribution Bell-shaped distribution

Select all that apply Which of the following is true regarding the graph depicting the normal probability density function f(x)? - Is often referred to as the normal curve - Is symmetric around the mean - Is not always symmetric around the mean - Is often referred to as the bell curve

- Is often referred to as the normal curve - Is symmetric around the mean - Is often referred to as the bell curve

Two events are _____ if the occurrence of one event does not affect the probability occurrence of another.

independent Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other event.

A simple probability distribution for a continuous random variable is called the:

Continuous uniform distribution

The Daytona 500 runs 40 race cars. Of the 40, 19 cars crashed. This is a probability of 0.475 that a car will crash in the race. This is an example of which probability?

empirical

The _____ of the discrete random variable X, denoted by E(X), or simply μ, is a weighted average of all possible values of X.

expected value - The expected value is also referred to as the mean.

Two events are '______' if the occurrence of one event does not affect the probability of the occurrence of the other event.

independent

A standard normal table, also referred to as the z-table, provides what information that is under the z curve?

probabilities

Which of the following are key properties of the discrete probability distribution?

- The probability of each value x is a value between 0 and 1, or, equivalently, 0 ≤ P(X = x) ≤ 1 - The sum of the probabilities equals 1. In other words, ΣP(X = xi) = 1, where the sum extends over all values x of X

He offers an annual bonus of $10,000 for superior performance, $6,000 for good performance, $3,000 for fair performance, and $0 for poor performance. Based on prior records, he has an expected value of the annual bonus of $4,000. What is the total annual amount that Brad can expect to pay in bonuses if he has 10 employees?

$40,000

A Bernoulli process consists of a series of n independent and identical trials of an experiment such that on each trial: (Choose all that apply!)

- The probabilities of success and failure remain the same from trial to trial - There are only two possible outcomes

Alison has been hired to sell two different homes on the same street that two houses apart. She predicts that Home A has a 61% chance in selling on the first week of being listed, whereas Home B is in lesser condition and has a 26% probability. There is also a 16% chance both homes will not sell on the first week of it being listed. What is the probability that Home A doesn't sell in the first week because of House B's lesser condition?

0.615 Use the conditional probability rule: P(A|B) = P(A ∩ B)P(B) = 0.160.26 = 0.615P(A ∩ B)P(B) = 0.160.26 = 0.615 .

In Holland, 30% of the people own a car. If five adults are randomly selected, what is the probability that two or more own a car?

83.7% probability that no more than two own a car. P (X = 0) = 5!0!(5−0)! × (0.30)0 × (0.70)5−0 =0 .1681 P (X = 1) = 5!1!(5-1)! × (0.30)1 × (0.70)5-1 = 0.3602 P (X = 2) = 5!2!(5−2)! × (0.30)2 × (0.70)5−2 = 0.3087 P (X ≤ 2) = 0.1681 + 0.3602 + 0.3087 = 0.8370 or 83.7% likelihood no more than two will have a car.

Alex has been studying for the certified management exam. Results from the last exam indicate that the mean was 62 with a standard deviation of 7. He needs to be in the top 20% (80th percentile) to pass. The z table indicates 1.28. What score will place Alex in the top 20% of the distribution?

70.96 μ = 62, σ = 7, z = 1.28; x = 62 + 1.28(7) = 70.96.

What type of variable assumes a countable number of distinct values such as x1, x2, x3, and so on?

Discrete

What is the most widely used continuous probability distribution? the '______' distribution.

Normal

Subjective probability

The subjective probability is based on an individual's personal judgment or experience.

Conditional probability

The total probability rule is a useful tool for breaking the computation of a probability into distinct cases

The union of two events is denoted as

A ∪ B.

What do we refer to events which include all outcomes in the sample space?

ANSWER: Exhaustive - Mutually exclusive -Reason: Another important probability concept concerns mutually exclusive events. For two mutually exclusive events, the occurrence of one event precludes the occurrence of the other. - Inclusive - Reason: Well, that is a word but not what we are looking for! - Simple - Reason: An event is any subset of outcomes of the experiment. It is called a simple event if it contains a single outcome.

What is the probability theory rule that is a tool for breaking the computation of a probability into distinct cases?

ANSWER: Total probability rule Incorrect option: - Bayes' Theorem - Bayes' theorem uses this rule to update the probability of an event that has been affected by a new piece of evidence

Are the following examples; the return on a mutual fund, time to completion of a task, or the volume of beer sold as 16 ounces, examples of continuous or discrete random variables?

Continuous

Which of the following are the two defining properties of probability? - The empirical probability of an event is the observed relative frequency with which an event occurs - The probability of any event A is a value between 0 and 1; that is, 0 ≤ P(A) ≤ 1. - The sum of the probabilities of any list of mutually exclusive and exhaustive events equals 1. - The subjective probability is based on an individual's personal judgment or experience

ANSWER: - The probability of any event A is a value between 0 and 1; that is, 0 ≤ P(A) ≤ 1. - The sum of the probabilities of any list of mutually exclusive and exhaustive events equals 1. Incorrect answers but still true statements: - The empirical probability of an event is the observed relative frequency with which an event occurs - The subjective probability is based on an individual's personal judgment or experience

Random variables can also be defined in terms of their cumulative distribution function, or, equivalently, P(X ? x). What is the correct mathematical sign (instead of the ?) in the P(X ? x) for the cumulative distribution function?

≤ (less than or equal)

Johnny feels that he has a 85% chance of getting an A in Marketing and a 45% chance of getting an A in Managerial Economics. He also believes he has a 35% chance of getting an A in both classes. What is the probability that he does not get an A in either of these courses?

ANSWER: .05 Solution: 1 - (0.85+0.45−0.35)=0.05

Classical probabilities

Classical probabilities are often used in games of chance. They are based on the assumption that all outcomes of an experiment are equally likely.

Which is not a characteristic of the normal distribution?

It is inverse. Normal distribution is bell-shaped, symmetric, and asymptotic. Inverse is the inverse transformation converting Z to X to produce a corresponding value.

A manager believes that 20% of consumers will respond positively to the firm's social media campaign. Also, 24% of those who respond positively will become loyal customers.Find the probability that the next recipient of their social media campaign will react positively and will become a loyal customer?

P(R ∩ L) =P(L∣R)P(R) = 0.24 × 0.20 =.048

Scores on a management aptitude exam are normally distributed with a mean of 72 and a standard deviation of 8. If we are trying to find the probability that a randomly selected manager will score above 75, what is the corresponding Z value?

P(Z >.375) Reason: P(Z>(75−72)/8 = P(Z >.375)

Statistical analysis

The total probability rule is a useful tool for breaking the computation of a probability into distinct cases

He offers an annual bonus of $10,000 for superior performance, $6,000 for good performance, $3,000 for fair performance, and $0 for poor performance. Based on prior records, he expects an employee to perform at superior, good, fair, and poor performance levels with probabilities 0.10, 0.20, 0.50, and 0.20, respectively. Calculate the expected value of the annual bonus amount

$3,700 Reason: =10,000*.1+(6,000*.2) +3,000(.5)+(0).2 = $3,700

We cannot describe the possible values of a '_______' random variable X with a list x1, x2,... because the value (x1 + x2)/2, not in the list, might also be possible

'Continuous' random variable

For a Poisson process, we define the number of '________' achieved in a specified time or space interval as a Poisson random variable.

'successes'

An experiment satisfies a Poisson process if (choose all that apply)

- The probability of success in any interval is the same for all intervals of equal size - The probability of success in any interval is proportional to the size of the interval - The number of successes within a specified time or space interval equals any integer between zero and infinity

For the binomial distribution, px(1 − p)n − x, represents the probability of any particular sequence with x successes and n − x failures. Use this formula to answer the following: In the Southern area of the United States, approximately 20% of adults have a college degree. We randomly ask four adults whether they have a college degree. Which of the following statements is true?

- n=4 - Probability that one adult will have a college degree = 10.24%

For the binomial distribution, px(1 − p)n − x, represents the probability of any particular sequence with x successes and n − x failures. Use this formula to answer the following: In the Southern area of the United States, approximately 20% of adults have a college degree. We randomly ask four adults whether they have a college degree. What is the probability that none of the adults have a college degree?

.4096

In Holland, 74% of the people own a car. If five adults are randomly selected, what is the probability that none of the five have a car?

0.12% Using the Bernoulli process, the probability of success (having a car) is p = 0.74 and the probability of failure (not having a car) is 1 − p = 1 − 0.74 = 0.26. The probability of none of the five people having a car is x = 0 thus: P (X = 0)= 5!0!(5 − 0)! × (0.74)0 × (0.26)5−0 = 0.00119.

In reviewing retirement portfolios, Kim determined the probability of a client owning stock is 0.70 and the probability of owning a bond is 0.20. The probability of a customer who owns bonds already owning stock is 0.60. What is the probability a client owns both securities in their retirement portfolio?

0.42 Use the multiplication rule (joint probability); P(S ∩ B) = P(B|S)P(S) = 0.60 × 0.70 = 0.420.

Michael has interviewed for two jobs. He feels that he has a 60% chance of getting an offer on Job A and a 55% chance of getting an offer on Job B. He also believes there is a 40% chance of getting an offer on both jobs. What is the probability that he receives an offer on at least one of the jobs?

0.75 P (A) = 0.60; P (B) = 0.55; P (A ∩ B) = 0.40. Use the addition rule. P (A ∪ B) = 0.60 + 0.55 − 0.40 = 0.75.

In Holland, 60% of the people own a car. If five adults are randomly selected, what is the probability that none of the five have a car?

1.02% Using the Bernoulli process, the probability of success (having a car) is p = 0.60 and the probability of failure (not having a car) is 1 − p = 1 − 0.60 = 0.40. The probability of none of the five people having a car is x = 0 thus: P (X = 0)= 5!0!(5 − 0)! × (0.60)0 × (0.40)5−0 = 0.01024P X = 0= 5!0!(5 - 0)! × 0.600 × 0.405-0 = 0.01024 .

Select all that apply Scores on a management aptitude examination are normally distributed with a mean of 72 and a standard deviation of 8. We want to find the lowest score that will place a manager in the top 10% (90th percentile) of the distribution. Which of the following is true to solve this problem? - The 90th percentile is a numerical value x such that P(X < x) = 0.90 - a score of 82.24 or higher will place a manager in the top 10% of the distribution - z = 1.28 - We will use the inverse transformation x + μ = zσ to solve these problems.

- The 90th percentile is a numerical value x such that P(X < x) = 0.90 - a score of 82.24 or higher will place a manager in the top 10% of the distribution - z = 1.28

A special case where the mean is equal to zero and the variance is equal to one is called _____.

standard normal distribution The standard normal distribution is a special case of the normal distribution with a mean equal to zero and a standard deviation (or variance) equal to one.