AIM 1: Define random variables, and distinguish between continuous and discrete random variables.

1、Which of the following statements about probability is most accurate?

A) A conditional probability is the probability that two or more events will happen concurrently.

B) An outcome is the calculated probability of an event.

C) Out of a sample of 100 widgets 10 were found to be defective, 20 were perfect, and 70 were OK. The probability of picking a perfect widget at random is 29%.

D) An event is a set of one or more possible values of a random variable.

The correct answer is D

Conditional probability is the probability of one event happening given that another event has happened. An outcome is the numerical result associated with a random variable. The probability of picking a perfect widget is 20 / 100 = 0.20 or 20%.

2、Let A and B be two mutually exclusive events with P(A) = 0.40 and P(B) = 0.20. Therefore:fficeffice" />

A) P(B|A) = 0.20.

B) P(A or B) = 0.52.

C) P(A and B) = 0.

D) P(A and B) = 0.08.

The correct answer is C

If the two evens are mutually exclusive, the probability of both ocurring is zero.

AIM 2: Define the probability of event.

1、The following table summarizes the availability of trucks with air bags and bucket seats at a dealership.

What is the probability of randomly selecting a truck with air bags and bucket seats?

A) 0.34.

B) 0.16.

C) 0.28.

D) 0.57.

The correct answer is A

75 / 220 = 0.34.

2、If two fair coins are flipped and two fair six-sided dice are rolled, all at the same time, what is the probability of ending up with two heads (on the coins) and two sixes (on the dice)?

A) 0.4167.

B) 0.0069.

C) 0.0039.

D) 0.8333.

The correct answer is B

3、A dealer in a casino has rolled a five on a single die three times in a row. What is the probability of her rolling another five on the next roll, assuming it is a fair die?

A) 0.001.

B) 0.167.

C) 0.500.

D) 0.200.

The correct answer is B

The probability of a value being rolled is 1/6 regardless of the previous value rolled.

4、Given the following table about employees of a company based on whether they are smokers or nonsmokers and whether or not they suffer from any allergies, what is the probability of both suffering from allergies and not suffering from allergies?

A)   0.50.

B)   0.00.

C)   1.00.

D)   0.24.

The correct answer is B

These are mutually exclusive, so the joint probability is zero.

5、The probability of a new Wal-Mart being built in town is 64%. If Wal-Mart comes to town, the probability of a new Wendy’s restaurant being built is 90%. What is the probability of a new Wal-Mart and a new Wendy’s restaurant being built?

A) 0.576.

B) 0.675.

C) 0.240.

D) 0.306.

The correct answer is A

P(AB) = P(A|B) × P(B)

The probability of a new Wal-Mart and a new Wendy’s is equal to the probability of a new Wendy’s “if Wal-Mart” (0.90) times the probability of a new Wal-Mart (0.64). (0.90)(0.64) = 0.576.

6、A two-sided but very thick coin is expected to land on its edge twice out of every 100 flips. And the probability of face up (heads) and the probability of face down (tails) are equal. When the coin is flipped, the prize is $1 for heads, $2 for tails, and $50 when the coin lands on its edge. What is the expected value of the prize on a single coin toss?

A) $17.67.

B) $2.47.

C) $1.50.

D) $26.50.

The correct answer is B

Since the probability of the coin landing on its edge is 0.02, the probability of each of the other two events is 0.49. The expected payoff is: (0.02 × $50) + (0.49 × $1) + (0.49 × $2) = $2.47.

7、The following table summarizes the availability of trucks with air bags and bucket seats at a dealership.

What is the probability of selecting a truck at random that has either air bags or bucket seats?

A)    34%.

B)    73%.

C)   50%.

D)   107%.

The correct answer is Bfficeffice" />

The addition rule for probabilities is used to determine the probability of at least one event among two or more events occurring. The probability of each event is added and the joint probability (if the events are not mutually exclusive) is subtracted to arrive at the solution. P(air bags or bucket seats) = P(air bags) + P(bucket seats) ? P(air bags and bucket seats) = (125 / 220) + (110 / 220) ? (75 / 220) = 0.57 + 0.50 ? 0.34 = 0.73 or 73%.

Alternative: 1 ? P(no airbag and no bucket seats) = 1 ? (60 / 220) = 72.7%

8、If the probability of both a new Wal-Mart and a new Wendy’s being built next month is 68% and the probability of a new Wal-Mart being built is 85%, what is the probability of a new Wendy’s being built if a new Wal-Mart is built?

A) 0.70.

B) 0.80.

C) 0.60.

D) 0.85.

The correct answer is B

P(AB) = P(A|B) × P(B)

0.68 / 0.85 = 0.80

9、Thomas Baynes has applied to both Harvard and Yale. Baynes has determined that the probability of getting into Harvard is 25% and the probability of getting into Yale (his father’s alma mater) is 42%. Baynes has also determined that the probability of being accepted at both schools is 2.8%. What is the probability of Baynes being accepted at either Harvard or Yale?

A) 64.2%.

B) 7.7%.

C) 10.5%.

D) 67.0%.

The correct answer is B

Using the addition rule, the probability of being accepted at Harvard or Yale, is equal to: P(Harvard) + P(Yale) ? P(Harvard and Yale) = 0.25 + 0.42 ? 0.028 = 0.642 or 64.2%.

10、A joint probability of A and B must always be:

A) greater than or equal to the conditional probability of A given B.

B) greater than or equal to than the probability of A or B.

C) less than or equal to the conditional probability of A given B.

D) less than the probability of A and the probability of B.

The correct answer is C

By the formula for joint probability: P(AB)=P(A|B) × P(B), since P(B) ≤ 1, then P(AB) ≤P(A|B). None of the other choices must hold.

11、A conditional expectation involves:

A) refining a forecast because of the occurrence of some other event.

B) determining the expected joint probability.

C) calculating the conditional variance.

D) estimating the skewness.

The correct answer is A

Conditional expected values are contingent upon the occurrence of some other event. The expectation changes as new information is revealed.

12、An investor has an A-rated bond, a BB-rated bond, and a CCC-rated bond where the probabilities of default over the next three years are 4 percent, 12 percent, and 30 percent, respectively. What is the probability that all of these bonds will default in the next three years if the individual default probabilities are independent?

A) 1.44%.

B) 23.00%.

C) 0.14%.

D) 46.00%.

The correct answer is C

Since the probability of default for each bond is independent, P(ABBCCC) = P(A) × P(BB) × P(CCC) = 0.04 × 0.12 × 0.30 = 0.00144 = 0.14%.

13、If a fair coin is tossed twice, what is the probability of obtaining heads both times?

A) 1/2.

B) 3/4. 

C) 1/4.

D) 1.

The correct answer is C

The probability of tossing a head, H, is P(H) = 1/2. Since these are independent events, the probability of two heads in a row is P(HH) = P(H) × P(H) = 1/2 × 1/2 = 1/4.

14、Dependent random variables are defined as variables where their joint probability is:

A) equal to zero.

B) greater than the product of their individual probabilities.

C) not equal to the product of their individual probabilities.

D) equal to the product of their individual probabilities.

The correct answer is C

Dependence results between random variables when their joint probabilities are not equal products of individual probabilities. If they are equal, then an independent relationship exists.

15、An investor is choosing one of twenty securities. Ten of the securities are stocks and ten are bonds. Four of the ten stocks were issued by utilities, the other six were issued by industrial firms. Two of the ten bonds were issued by utilities, the other eight were issued by industrial firms. If the investor chooses a security at random, the probability that it is a bond or a security issued by an industrial firm is:

A) 0.80.

B) 0.70.

C) 0.60.

D) 0.50.

The correct answer is A

Let B represent the set of bonds and I the set of industrial firms. The desired probability is the probability of the union of sets B and I, P(B I). According to the theorems of probability, P(B I)=P(B)+P(I)-P(B∩I), where P(B) is the probability that a security is a bond=P(B)=10/20, P(I) is the probability that a security was issued by an industrial firm=P(I)=14/20, and P(B∩I) is the probability that a security is both a bond and issued by an industrial firm=P(B∩I)=8/20. 10/20+14/20-8/20=16/20=0.80.

16、X and Y are discrete random variables. The probability that X = 3 is 0.20 and the probability that Y = 4 is 0.30. The probability of observing that X = 3 and Y = 4 concurrently is closest to:

A) 0.

B) Cannot answer with the information provided.

C) 0.06.

D) 0.50.

The correct answer is B

If you knew that X and Y were independent, you could calculate the probability as 0.20(0.30) = 0.06. Without this knowledge, you would need the joint probability distribution.

17、The probabilities that three students will earn an A on an exam are 0.20, 0.25, and 0.30, respectively. If each student’s performance is independent of that of the other two students, the probability that all three students will earn an A is closest to:

A) 0.0150.

B) 0.0075.

C) 0.0010.

D) 0.7500.

The correct answer is A

Since the events are independent, the probability is obtained by multiplying the individual probabilities. Probability of three As = (0.20) × (0.25) × (0.30) = 0.0150.

AIM 3: Explain the relative frequency or the empirical definition of probability.

1、

The number of intervals in this frequency table is:

A)    5.

B)    1.

C)   16.

D)   50.

The correct answer is A

An interval is the set of return values that an observation falls within. Simply count the return intervals on the table—there are 5 of them.

[此贴子已经被作者于2009-6-25 9:22:34编辑过]

The sample size is:

A) 1.

B) 16.

C) 5.

D) 50.

The correct answer is B

The sample size is the sum of all of the frequencies in the distribution, or 3 + 7 + 3 + 2 + 1 = 16.

The relative frequency of the second class is:

A) 10.0%.

B) 0.0%.

C) 16.0%.

D) 43.8%.

The correct answer is D

The relative frequency is found by dividing the frequency of the interval by the total number of frequencies:7/16=43.8%

2、How is the relative frequency of an interval computed?

A) By dividing the sum of the two interval limits by 2.

B) By dividing the frequency of that interval by the sum of all frequencies.

C) By multiplying the frequency of the interval by 100.

D) By subtracting the lower limit of the interval by the upper limit.

The correct answer is B

The relative frequency is the percentage of total observations falling within each interval. It is found by taking the frequency of the interval and dividing that number by the sum of all frequencies.

AIM 4: Explain Bayes’ theorem and use Bayes’ formula to determine the probability of causes for a given event.

1、Bonds rated B have a 25% chance of default in five years. Bonds rated CCC have a 40% chance of default in five years. A portfolio consists of 30% B and 70% CCC-rated bonds. If a randomly selected bond defaults in a five-year period, what is the probability that it was a B-rated bond?

A) 0.625.

B) 0.429.

C) 0.211.

D) 0.250.

The correct answer is C

According to Bayes' formula: P(B/default) = P(default and B)/P(default).

P(default and B )= P(default/B) × P(B) = 0.250 × 0.300 = 0.075

P(default and CCC) = P(default/CCC) × P(CCC) = 0.400 × 0.700 = 0.280

P(default) = P(default and B) + P(default and CCC) = 0.355

P(B/default) = P(default and B)/P(default) = 0.075 / 0.355 = 0.211

2、The probability of A is 0.4. The probability of AC is 0.6. The probability of (B|A) is 0.5, and the probability of (B|AC) is 0.2. Using Bayes’ formula, what is the probability of (A|B)?

A) 0.875.

B) 0.125.

C) 0.375.

D) 0.625.

The correct answer is D

Using the total probability rule, we can compute the P(B):

P(B) = [P(B|A) × P(A)] + [P(B|AC) × P(AC)]

P(B) = [0.5 × 0.4] + [0.2 × 0.6] = 0.32

Using Bayes’ formula, we can solve for P(A|B):

P(A|B) = [ P(B|A) / P(B) ] × P(A) = [0.5 / 0.32] × 0.4 = 0.625

3、A major securities exchange is considering the introduction of a new derivative contract. In the past, the success rate for new derivatives has been 30 percent. Extensive market research has produced a positive marketing research report for the contract under consideration. Historically, 70 percent of the successful contracts have received favorable reports prior to introduction. Only 10 percent of unsuccessful contracts have received favorable reports. What is the probability that the new contract will be successful?

A) 87%.

B) 75%.

C) 13%.

D) 25%.

The correct answer is B

Let S denote success, U denote unsuccessful, and F denote a favorable report. We have P(S)=0.30, P(U)=0.70, P(F | S)=0.70, and P(F | U)=0.10. Using Bayes theorem, the probability of a successful contract, given a favorable report is:

AIM 5: Define probability mass function, probability function, probability density function, cumulative distribution function.

Which of the following statements about probability distributions is least accurate?

A) In a binomial distribution each observation has only two possible outcomes that are mutually exclusive.

B) A probability distribution is, by definition, normally distributed.

C) A probability distribution includes a listing of all the possible outcomes of an experiment.

D) One of the key properties of a probability function is  0 ≤ p ≤ 1.

The correct answer is B

Probabilities must be zero or positive, but a probability distribution is not necessarily normally distributed. Binomial distributions are either successes or failures.

AIM 6: Discuss the implications the normal probability density function has.

1、A normal distribution can be completely described by its:

A) skewness and kurtosis.

B) mean and mode. 

C) mean and variance. 

D) standard deviation.

The correct answer is C

The normal distribution can be completely described by its mean and variance.

2、The lower limit of a normal distribution is:

A) negative infinity.

B) zero.

C) one.

D) negative one.

The correct answer is A

By definition, a true normal distribution has a positive probability density function from negative to positive infinity.

3、Which of the following statements about a normal distribution is least accurate?

A) A normal distribution has excess kurtosis of three.

B) The mean, median, and mode are equal.

C) The mean and variance completely define a normal distribution.

D) Approximately 68% of the observations lie within +/- 1 standard deviation of the mean.

The correct answer is A

Even though normal curves have different sizes, they all have identical shape characteristics. The kurtosis for all normal distributions is three; an excess kurtosis of three would indicate a leptokurtic distribution. The other choices are true.

AIM 7: Distinguish between univariate and multivariate probability density functions.

1、Which of the following would least likely be categorized as a multivariate distribution?

A) The days a stock traded and the days it did not trade.

B) The return of a stock and the return of the DJIA.

C) The days a stock traded and the days the DJIA went up.

D) The returns of the stocks in the DJIA.

The correct answer is A

2、A multivariate distribution:

A) gives multiple probabilities for the same outcome.

B) applies only to normal distributions.

C) applies only to binomial distributions.

D) specifies the probabilities associated with groups of random variables.

The correct answer is D

This is the definition of a multivariate distribution.

3、In a multivariate normal distribution, a correlation tells the:

A) strength of the linear relationship between two of the variables.

B) overall relationship between all the variables.

C) relationship between the means and variances of the variables.

D) relationship between the means and standard deviations of the variables.

The correct answer is A

This is true by definition. The correlation only applies to two variables at a time.

AIM 9: Explain the difference between statistical independence and statistical dependence.

1、If X and Y are independent events, which of the following is most accurate?

A) P(X or Y) = P(X) + P(Y).

B) X and Y cannot occur together.

C) P(X | Y) = P(X).

D) P(X or Y) = (P(X)) × (P(Y)).

The correct answer is C

Note that events being independent means that they have no influence on each other. It does not necessarily mean that they are mutually exclusive. Accordingly, P(X or Y) = P(X) + P(Y) ? P(X and Y). By the definition of independent events, P(X|Y) = P(X).

2、A company says that whether its earnings increase depends on whether it increased its dividends. From this we know:

A) P(both dividend increase and earnings increase) = P(dividend increase).  

B) P(earnings increase | dividend increase) is not equal to P(earnings increase). 

C) P(dividend increase or earnings increase) = P(both dividend and earnings increase). 

D) P(dividend increase | earnings increase) is not equal to P(earnings increase). 

The correct answer is B

If two events A and B are dependent, then the conditional probabilities of P(A|B) and P(B|A) will not equal their respective unconditional probabilities (of P(A) and P(B), respectively). The other choices may or may not occur, e.g., P(A | B) = P(B) is possible but not necessary.

3、If the outcome of event A is not affected by event B, then events A and B are said to be:

A) mutually exclusive.

B) statistically independent.

C) collectively exhaustive.

D) conditionally dependent.

The correct answer is B

If the outcome of one event does not influence the outcome of another, then the events are independent.

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