8. Probability Concepts (Web + Sch Note)

Joint Probabilities Returns Rb=.5 Rb=.2 Rb=-.3 Ra=-.1 .4 0 0 Ra=.1 0 .3 0 Ra=.3 0 0 .3 1. Given the joint probability table, the expected return of Stock A is closest to: A. 0.08. B. 0.12. C. 0.15. 2. Given the joint probability table, the standard deviation ofStock B is closest to: A. 0.11. B. 0.22. C. 0.33. 3. Given the joint probability table, the variance of Stock A is closest to: A. 0.03. B. 0.12. C. 0.17. 4. Given the joint probability table, the covariance between A and B is closest to: A. -0.160. B. -0.055. C. 0.004. 5. Given the joint probability table, the correlation between Ra and Rb is closest to: A. -0.99. B. 0.02. C. 0.86.

1. A Expected return of Stock A = (0.4)(—0.1) + (0.3)(0.1) + (0.3)(0.3) = 0.08 2. C Expected return of Stock B = (0.4) (0.5) + (0.3)(0.2) + (0.3)(—0.3) = 0.17 Var(Rb) = 0.4(0.5 - 0.17)^2 + 0.3(0.2- 0.17)^2 + 0.3(—0.3- 0.17)^2 = 0.1101 Standard deviation = (0.1101)^.5 = 0.3318 3. A E(Ra) = 0.08 (from above) Var(RA) = 0.4(-0.1 - 0.08)^2 + 0.3(0.1 - 0.08)^2 + 0.3(0.3- 0.08)^2 = 0.0276 4. B COV(Ra,Rb) = 0.4(-0.1 - 0.08)(0.5 - 0.17) + 0.3(0.1 - 0.08)(0.2- 0.17) + 0.3(0.3- 0.08)( -0.3- 0.17) = -0.0546 5. A Corr(RA,RB) = Cov(RA,RB) / stdev(RA)stdev(RB) Corr(RA,RB) = -0.0546 / (0.1661 )(0.3318) = -0.9907

Econ state Stock perf Good 0.30 Good 0.60 Neutral 0.30 Poor 0.10 Neutral 0.50 Good 0.30 Neutral 0.40 Poor 0.30 Poor 0.30 Good 0.10 Neutral 0.60 Poor 0.30 1. What is the conditional probability of having good stock performance in a poor economic environment? A. 0.02 B. 0.10 C. 0.30 2. What is the joint probability of having a good economy and a neutral stock performance? A. 0.09. B. 0.20. C. 0.30. 3. What is the total probability of having a good performance in the stock? A. 0.35. B. 0.65. C. 1.00. 4. Given that the stock had good performance, the probability the state of the economy was good is closest to: A. 0.35. B. 0.46. C. 0.51. 5. Consider a universe of ten bonds from which an investor will ultimately purchase six bonds for his portfolio. If the order in which he buys these bonds is not important, how many potential 6-bond combinations are there? A. 7. B. 210. C. 5,040.

1. B Go to the poor economic state and read off the probability of good performance [i.e., P(good performance| poor economy) = 0.10]. 2. A P(good economy and neutral performance) = P(good economy)P(neutral performan good economy) = (0.3)(0.3) = 0.09. 3. A (0.3)(0.6) + (0.5)(0.3) + (0.2)(0.1) = 0.35. This is the sum of all the joint probabilities for good performance over all states [i.e., EP(economic state;) P(good performance| economic state;)]. 4. C This is an application of Bayes' formula. P(good economy| good performance) = P(good stock performance| good economy) x P(good economy) / P(good stock performance). [ .6 * .3 ] / [ .3*.6 + .5*.3 + .2*.1 ] = .18/.35 = .5143 5. B Use calc: 10C6

The events Y and Z are mutually exclusive and exhaustive: P(Y) = 0.4 and P(Z) = 0.6. If the probability of X given Y is 0.9, and the probability of X given Z is 0.1, what is the unconditional probability of X? A) 0.33. B) 0.42. C) 0.40.

B Because the events are mutually exclusive and exhaustive, the unconditional probability is obtained by taking the sum of the two joint probabilities: P(X) = P(X | Y) × P(Y) + P(X | Z) × P(Z) = 0.4 × 0.9 + 0.6 × 0.1 = 0.42.

The covariance: A) must be between -1 and +1. B) can be positive or negative. C) must be positive.

B Cov(a,b) = σaσbρa,b. Since ρa,b can be positive or negative, Cov(a,b) can be positive or negative.

If a firm is going to create three teams of four from twelve employees. Which approach is the most appropriate for determining how the twelve employees can be selected for the three teams? A) Permutation formula. B) Labeling formula. C) Combination formula.

B This problem is a labeling problem where the 12 employees will be assigned one of three labels. It requires the labeling formula. In this case there are [(12!) / (4!4!4!)] = 34,650 ways to group the employees.

Data shows that 75 out of 100 tourists who visit New York City visit the Empire State Building. It rains or snows in New York City one day in five. What is the joint probability that a randomly choosen tourist visits the Empire State Building on a day when it neither rains nor snows? A) 95%. B) 60%. C) 15%.

B A joint probability is the probability that two events occur when neither is certain or a given. Joint probability is calculated by multiplying the probability of each event together. (0.75) × (0.80) = 0.60 or 60%.

For the task of arranging a given number of items without any sub-groups, this would require: A) the labeling formula. B) only the factorial function. C) the permutation formula.

B The factorial function, denoted n!, tells how many different ways n items can be arranged where all the items are included.

In any given year, the chance of a good year is 40%, an average year is 35%, and the chance of a bad year is 25%. What is the probability of having two good years in a row? A) 10.00%. B) 16.00%. C) 8.75%.

B The joint probability of independent events is obtained by multiplying the probabilities of the individual events together: (0.40) × (0.40) = 0.16 or 16%.

Which of the following statements is least accurate regarding covariance? A) Covariance can only apply to two variables at a time. B) Covariance can exceed one. C) A covariance of zero rules out any relationship.

C A covariance only measures the linear relationship. The covariance can be zero while a non-linear relationship exists. Both remaining statements are true.

An analyst expects that 20% of all publicly traded companies will experience a decline in earnings next year. The analyst has developed a ratio to help forecast this decline. If the company is headed for a decline, there is a 90% chance that this ratio will be negative. If the company is not headed for a decline, there is only a 10% chance that the ratio will be negative. The analyst randomly selects a company with a negative ratio. Based on Bayes' theorem, the updated probability that the company will experience a decline is: A) 26%. B) 18%. C) 69%.

C Given a set of prior probabilities for an event of interest, Bayes' formula is used to update the probability of the event, in this case that the company we have already selected will experience a decline in earnings next year. Bayes' formula says to divide the Probability of New Information given Event by the Unconditional Probability of New Information and multiply that result by the Prior Probability of the Event. In this case, P(company having a decline in earnings next year) = 0.20 is divided by 0.26 (which is the Unconditional Probability that a company having an earnings decline will have a negative ratio (90% have negative ratios of the 20% which have earnings declines) plus (10% have negative ratios of the 80% which do not have earnings declines) or ((0.90) × (0.20)) + ((0.10) × (0.80)) = 0.26.) This result is then multiplied by the Prior Probability of the ratio being negative, 0.90. The result is (0.20 / 0.26) × (0.90) = 0.69 or 69%.

Given P(X = 20, Y = 0) = 0.4, and P(X = 30, Y = 50) = 0.6, then COV(XY) is: A) 125.00. B) 25.00. C) 120.00.

C The expected values are: E(X) = (0.4 × 20) + (0.6 × 30) = 26, and E(Y) = (0.4 × 0) + (0.6 × 50) = 30. The covariance is COV(XY) = (0.4 × ((20 ? 26) × (0 ? 30))) + ((0.6 × (30 ? 26) × (50 ? 30))) = 120.

For a given corporation, which of the following is an example of a conditional probability? The probability the corporation's: A) earnings increase and dividend increases. B) inventory improves. C) dividend increases given its earnings increase.

C A conditional probability involves two events. One of the events is a given, and the probability of the other event depends upon that given.

There is a 30% chance that the economy will be good and a 70% chance that it will be bad. If the economy is good, your returns will be 20% and if the economy is bad, your returns will be 10%. What is your expected return? A) 13%. B) 17%. C) 15%.

A Expected value is the probability weighted average of the possible outcomes of the random variable. The expected return is: ((0.3) × (0.2)) + ((0.7) × (0.1)) = (0.06) + (0.07) = 0.13.

If X and Y are independent events, which of the following is most accurate? A) P(X | Y) = P(X). B) P(X or Y) = (P(X)) × (P(Y)). C) P(X or Y) = P(X) + P(Y).

A 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).

For a stock, which of the following is least likely a random variable? Its: A) stock symbol. B) current ratio. C) most recent closing price.

A A random variable must be a number. Sometimes there is an obvious method for assigning a number, such as when the random variable is a number itself, like a P/E ratio. A stock symbol of a randomly selected stock could have a number assigned to it like the number of letters in the symbol. The symbol itself cannot be a random variable.

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) 0.211. B) 0.625. C) 0.250.

A 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

There is a 40% chance that the economy will be good next year and a 60% chance that it will be bad. If the economy is good, there is a 50 percent chance of a bull market, a 30% chance of a normal market, and a 20% chance of a bear market. If the economy is bad, there is a 20% chance of a bull market, a 30% chance of a normal market, and a 50% chance of a bear market. What is the probability of a bull market next year? A) 32%. B) 20%. C) 50%.

A Because a good economy and a bad economy are mutually exclusive, the probability of a bull market is the sum of the joint probabilities of (good economy and bull market) and (bad economy and bull market): ((0.40) × (0.50)) + ((0.60) × (0.20)) = 0.32 or 32%.

A conditional expectation involves: A) refining a forecast because of the occurrence of some other event. B) calculating the conditional variance. C) determining the expected joint probability.

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

The correlation of returns between Stocks A and B is 0.50. The covariance between these two securities is 0.0043, and the standard deviation of the return of Stock B is 26%. The variance of returns for Stock A is: A. 0.0011. B. 0.0331. C. 0.2656.

A Corr(RA,RB) = Cov(RA,RB) / stdev(RA)stdev(RB) .5 = .0043 / stdev(RA)(.26) stdev(RB) = .0331 Var(RB) = .0011

Last year, the average salary increase for poultry research assistants was 2.5%. Of the 10,000 poultry research assistants, 2,000 received raises in excess of this amount. The odds that a randomly selected poultry research assistant received a salary increase in excess of 2.5% are: A) 1 to 4. B) 20%. C) 1 to 5.

A For event "E," the probability stated as odds is: P(E) / [1 - P(E)]. Here, the probability that a poultry research assistant received a salary increase in excess of 2.5% = 2,000 / 10,000 = 0.20, or 1/5 and the odds are (1/5) / [1 - (1/5)] = 1/4, or 1 to 4.

If the odds against an event occurring are twelve to one, what is the probability that it will occur? A) 0.0769. B) 0.0833. C) 0.9231.

A If the probability against the event occurring is twelve to one, this means that in thirteen occurrences of the event, it is expected that it will occur once and not occur twelve times. The probability that the event will occur is then: 1/13 = 0.0769.

Let A and B be two mutually exclusive events with P(A) = 0.40 and P(B) = 0.20. Therefore: A) P(A and B) = 0. B) P(B|A) = 0.20. C) P(A and B) = 0.08.

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

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) 0.306.

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.

If the probability of an event is 0.10, what are the odds for the event occurring? A) One to nine. B) One to ten. C) Nine to one.

A The answer can be determined by dividing the probability of the event by the probability that it will not occur: (1/10) / (9/10) = 1 to 9. The probability of the event occurring is one to nine, i.e. in ten occurrences of the event, it is expected that it will occur once and not occur nine times.

The covariance of returns on two investments over a 10-year period is 0.009. If the variance of returns for investment A is 0.020 and the variance of returns for investment B is 0.033, what is the correlation coefficient for the returns? A) 0.350. B) 0.444. C) 0.687.

A The correlation coefficient is: Cov(A,B) / [(Std Dev A)(Std Dev B)] = 0.009 / [(√0.02)(√0.033)] = 0.350.

There is a 90% chance that the economy will be good next year and a 10% chance that it will be bad. If the economy is good, there is a 60% chance that XYZ Incorporated will have EPS of $4.00 and a 40% chance that their earnings will be $3.00. If the economy is bad, there is an 80% chance that XYZ Incorporated will have EPS of $2.00 and a 20% chance that their earnings will be $1.00. What is the firm's expected EPS? A) $3.42. B) $5.40. C) $2.50.

A The expected EPS is calculated by multiplying the probability of the economic environment by the probability of the particular EPS and the EPS in each case. The expected EPS in all four outcomes are then summed to arrive at the expected EPS: (0.90 × 0.60 × $4.00) + (0.90 × 0.40 × $3.00) + (0.10 × 0.80 × $2.00) + (0.10 × 0.20 × $1.00) = $2.16 + $1.08 + $0.16 + $0.02 = $3.42.

With respect to the units each is measured in, which of the following is the most easily directly applicable measure of dispersion? The: A) standard deviation. B) covariance. C) variance.

A The standard deviation is in the units of the random variable itself and not squared units like the variance. The covariance would be measured in the product of two units of measure.

Which of the following is an empirical probability? A) For a stock, based on prior patterns of up and down days, the probability of the stock having a down day tomorrow. B) The probability the Fed will lower interest rates prior to the end of the year. C) On a random draw, the probability of choosing a stock of a particular industry from the S& 500 based on the number of firms.

A There are three types of probabilities: a priori, empirical, and subjective. An empirical probability is calculated by analyzing past data.

A portfolio manager wants to eliminate four stocks from a portfolio that consists of six stocks. How many ways can the four stocks be sold when the order of the sales is important? A) 360. B) 180. C) 24.

A This is a choose four from six problem where order is important. Thus, it requires the permutation formula: n! / (n ? r)! = 6! / (6 ? 4)! = 360. With TI calculator: 6 [2nd][nPr] 4 = 360.

An analyst announces that an increase in the discount rate next quarter will double her earnings forecast for a firm. This is an example of a: A) conditional expectation. B) use of Bayes' formula. C) joint probability.

A This is a conditional expectation. The analyst indicates how an expected value will change given another event.

A firm wants to select a team of five from a group of ten employees. How many ways can the firm compose the team of five? A) 252. B) 120. C) 25.

A This is a labeling problem where there are only two labels: chosen and not chosen. Thus, the combination formula applies: 10! / (5! × 5!) = 3,628,800 / (120 × 120) = 252. With a TI calculator: 10 [2nd][nCr] 5 = 252.

After repeated experiments, the average of the outcomes should converge to: A) the expected value. B) the variance. C) one.

A This is the definition of the expected value. It is the long-run average of all outcomes.

John purchased 60% of the stocks in a portfolio, while Andrew purchased the other 40%. Half of John's stock-picks are considered good, while a fourth of Andrew's are considered to be good. If a randomly chosen stock is a good one, what is the probability John selected it? A) 0.75. B) 0.40. C) 0.30.

A Using the information of the stock being good, the probability is updated to a conditional probability: P(John | good) = P(good and John) / P(good). P(good and John) = P(good | John) × P(John) = 0.5 × 0.6 = 0.3. P(good and Andrew) = 0.25 × 0.40 = 0.10. P(good) = P(good and John) + P (good and Andrew) = 0.40. P(John | good) = P(good and John) / P(good) = 0.3 / 0.4 = 0.75.

A firm holds two $50 million bonds with call dates this week. The probability that Bond A will be called is 0.80. The probability that Bond B will be called is 0.30. The probability that at least one of the bonds will be called is closest to: A) 0.86. B) 0.24. C) 0.50.

A We calculate the probability that at least one of the bonds will be called using the addition rule for probabilities: P(A or B) = P(A) + P(B) - P(A and B), where P(A and B) = P(A) × P(B) P(A or B) = 0.80 + 0.30 - (0.8 × 0.3) = 0.86

The following information is available concerning expected return and standard deviation of Pluto and Neptune Corporations: E(r) Stdev Pluto Corporation 11% 0.22 Neptune Corporation 9% 0.13 If the correlation between Pluto and Neptune is 0.25, determine the expected return and standard deviation of a portfolio that consists of 65% Pluto Corporation stock and 35% Neptune Corporation stock. A) 10.3% expected return and 16.05% standard deviation. B) 10.3% expected return and 2.58% standard deviation. C) 10.0% expected return and 16.05% standard deviation.

A ERPort = (WPluto)(ERPluto) + (WNeptune)(ERNeptune) = (0.65)(0.11) + (0.35)(0.09) = 10.3% σp = [(w1)2(σ1)2 + (w2)2(σ2)2 + 2w1w2σ1σ2 r1,2]1/2 = [(0.65)2(22)2 + (0.35)2(13)2 + 2(0.65)(0.35)(22)(13)(0.25)]1/2 = [(0.4225)(484) + (0.1225)(169) + 2(0.65)(0.35)(22)(13)(0.25)]1/2 = (257.725)1/2 = 16.0538%

A parking lot has 100 red and blue cars in it. 40% of the cars are red. 70% of the red cars have radios. 80% of the blue cars have radios. What is the probability of selecting a car at random and having it be red and have a radio? A) 28%. B) 48%. C) 25%.

A Joint probability is the probability that both events, in this case a car being red and having a radio, happen at the same time. Joint probability is computed by multiplying the individual event probabilities together: P(red and radio) = (P(red)) × (P(radio)) = (0.4) × (0.7) = 0.28 or 28%.

Jay Hamilton, CFA, is analyzing Madison, Inc., a distressed firm. Hamilton believes the firm's survival over the next year depends on the state of the economy. Hamilton assigns probabilities to four economic growth scenarios and estimates the probability of bankruptcy for Madison under each: Prob of growth Prob of bankruptcy Recession (< 0%) 20% 60% Slow growth (0% to 2%) 30% 40% Normal growth (2% to 4%) 40% 20% Rapid growth (> 4%) 10% 10% Based on Hamilton's estimates, the probability that Madison, Inc. does not go bankrupt in the next year is closest to: A) 67%. B) 18%. C) 33%.

A Using the total probability rule, the unconditional probability of bankruptcy is (0.2)(0.6) + (0.3)(0.4) + (0.4)(0.2) + (0.1) (0.1) = 0.33. The probability that Madison, Inc. does not go bankrupt is 1 - 0.33 = 0.67 = 67%.

Among 900 taxpayers with incomes below $100,000, 35 were audited by the IRS. The probability that a randomly chosen individual with an income below $100,000 was audited is closest to: A. 0.039. B. 0.125. C. 0.350.

A 35 / 900 = 0.0389

The probability that the DJIA will increase tomorrow is 2/3. The probability of an increase in the DJIA stated as odds is: A. two-to-one. B. one-to-three. C. two-to-three.

A Odds for E = P(E) / [1 — P(E)] =- = 2 /1 = two-to-one

Given Cov(X,Y) = 1,000,000. What does this indicate about the relationship between X and Y? A) It is strong and positive. B) Only that it is positive. C) It is weak and positive.

B A positive covariance indicates a positive linear relationship but nothing else. The magnitude of the covariance by itself is not informative with respect to the strength of the relationship.

For an unconditional probability: A) there are at least two events. B) there is only one random variable of concern. C) the addition rule is important.

B An unconditional probability gives the probability of an event regardless of what other events occur.

Which of the following statements about probability is most accurate? A) An outcome is the calculated probability of an event. B) An event is a set of one or more possible values of a random variable. C) A conditional probability is the probability that two or more events will happen concurrently.

B 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.

Given the conditional probabilities in the table below and the unconditional probabilities P(Y = 1) = 0.3 and P(Y = 2) = 0.7, what is the expected value of X? xi P(xi\Y= 1) P(xi \ Y= 2) 0 0.2 0.1 5 0.4 0.8 A. 5.0 B. 5.3 C. 5.7

B E(X| Y = 1) = (0.2) (0) + (0.4)(5) + (0.4)(10) = 6 and E(X| Y = 2) = (0.1)(0) + (0.8)(5) + (0.1)(10) = 5 E(X) = (0.3)(6) + (0.7)(5) = 5.30

If the outcome of event A is not affected by event B, then events A and B are said to be: A) conditionally dependent. B) statistically independent. C) mutually exclusive.

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

A company says that whether it increases its dividends depends on whether its earnings increase. 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 | earnings increase) is not equal to P(earnings increase).

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). Both remaining choices may or may not occur, e.g., P(A | B) = P(B) is possible but not necessary.

There is a 40% chance that the economy will be good next year and a 60% chance that it will be bad. If the economy is good, there is a 50 percent chance of a bull market, a 30% chance of a normal market, and a 20% chance of a bear market. If the economy is bad, there is a 20% chance of a bull market, a 30% chance of a normal market, and a 50% chance of a bear market. What is the joint probability of a good economy and a bull market? A) 50%. B) 20%. C) 12%.

B Joint probability is the probability that both events, in this case the economy being good and the occurrence of a bull market, happen at the same time. Joint probability is computed by multiplying the individual event probabilities together: (0.40) × (0.50) = 0.20 or 20%.

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) 0.60. B) 0.80. C) 0.70.

B P(AB) = P(A|B) × P(B) 0.68 / 0.85 = 0.80

At a charity ball, 800 names are put into a hat. Four of the names are identical. On a random draw, what is the probability that one of these four names will be drawn? A. 0.004. B. 0.005. C. 0.010.

B P(name 1 or name 2 or name 3 or name 4) = 1/800 + 1/800 + 1/800 + 1/800 = 4/800 = 0.005

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) $1.50. B) $2.47. C) $17.67.

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.

If the probability of an event is 0.20, what are the odds against the event occurring? A) Five to one. B) Four to one. C) One to four.

B The answer can be determined by dividing the probability of the event by the probability that it will not occur: (1/5) / (4/5) = 1 to 4. The probability against the event occurring is four to one, i.e. in five occurrences of the event, it is expected that it will occur once and not occur four times.

If the probability of an event is 0.20, what are the odds against the event occurring? A) One to four. B) Four to one. C) Five to one.

B The answer can be determined by dividing the probability of the event by the probability that it will not occur: (1/5) / (4/5) = 1 to 4. The probability against the event occurring is four to one, i.e. in five occurrences of the event, it is expected that it will occur once and not occur four times.

The covariance of the returns on investments X and Y is 18.17. The standard deviation of returns on X is 7%, and the standard deviation of returns on Y is 4%. What is the value of the correlation coefficient for returns on investments X and Y? A) +0.85. B) +0.65. C) +0.32.

B The correlation coefficient = Cov (X,Y) / [(Std Dev. X)(Std. Dev. Y)] = 18.17 / 28 = 0.65

There is a 60% chance that the economy will be good next year and a 40% chance that it will be bad. If the economy is good, there is a 70% chance that XYZ Incorporated will have EPS of $5.00 and a 30% chance that their earnings will be $3.50. If the economy is bad, there is an 80% chance that XYZ Incorporated will have EPS of $1.50 and a 20% chance that their earnings will be $1.00. What is the firm's expected EPS? A) $5.95. B) $3.29. C) $2.75.

B The expected EPS is calculated by multiplying the probability of the economic environment by the probability of the particular EPS and the EPS in each case. The expected EPS in all four outcomes are then summed to arrive at the expected EPS: (0.60 × 0.70 × $5.00) + (0.60 × 0.30 × $3.50) + (0.40 × 0.80 × $1.50) + (0.40 × 0.20 × $1.00) = $2.10 + $0.63 + $0.48 + $0.08 = $3.29.

Given P(X = 2, Y = 10) = 0.3, P(X = 6, Y = 2.5) = 0.4, and P(X = 10, Y = 0) = 0.3, then COV(XY) is: A) 24.0. B) -12.0. C) 6.0.

B The expected values are: E(X) = (0.3 × 2) + (0.4 × 6) + (0.3 × 10) = 6 and E(Y) = (0.3 × 10.0) + (0.4 × 2.5) + (0.3 × 0.0) = 4. The covariance is COV(XY) = ((0.3 × ((2 ? 6) × (10 ? 4))) + ((0.4 × ((6 ? 6) × (2.5 ? 4))) + (0.3 × ((10 ? 6) × (0 ? 4))) = ?12.

A company has two machines that produce widgets. An older machine produces 16% defective widgets, while the new machine produces only 8% defective widgets. In addition, the new machine employs a superior production process such that it produces three times as many widgets as the older machine does. Given that a widget was produced by the new machine, what is the probability it is NOT defective? A) 0.76. B) 0.92. C) 0.06.

B The problem is just asking for the conditional probability of a defective widget given that it was produced by the new machine. Since the widget was produced by the new machine and not selected from the output randomly (if randomly selected, you would not know which machine produced the widget), we know there is an 8% chance it is defective. Hence, the probability it is not defective is the complement, 1 - 8% = 92%.

Assume two stocks are perfectly negatively correlated. Stock A has a standard deviation of 10.2% and stock B has a standard deviation of 13.9%. What is the standard deviation of the portfolio if 75% is invested in A and 25% in B? A) 0.00%. B) 4.18%. C) 0.17%.

B The standard deviation of the portfolio is found by: [W12 σ12 + W22 σ22 + 2W1W2σ1σ2r1,2]0.5, or [(0.75)2(0.102)2 + (0.25)2(0.139)2 + (2)(0.75)(0.25)(0.102)(0.139)(-1.0)]0.5 = 0.0418, or 4.18%.

Compute the standard deviation of a two-stock portfolio if stock A (40% weight) has a variance of 0.0015, stock B (60% weight) has a variance of 0.0021, and the correlation coefficient for the two stocks is -0.35? A) 1.39%. B) 2.64%. C) 0.07%.

B The standard deviation of the portfolio is found by: [W12σ12 + W22σ2 2+ 2W1W2σ1σ2ρ1,2]0.5 = [(0.40)2(0.0015) + (0.60)2 (0.0021) + (2)(0.40)(0.60)(0.0387)(0.0458)(-0.35)]0.5 = 0.0264, or 2.64%.

There is a 50% chance that the Fed will cut interest rates tomorrow. On any given day, there is a 67% chance the DJIA will increase. On days the Fed cuts interest rates, the probability the DJIA will go up is 90%. What is the probability that tomorrow the Fed will cut interest rates or the DJIA will go up? A) 0.33. B) 0.72. C) 0.95.

B This requires the addition formula. From the information: P(cut interest rates) = 0.50 and P(DJIA increase) = 0.67, P(DJIA increase | cut interest rates) = 0.90. The joint probability is 0.50 × 0.90 = 0.45. Thus P (cut interest rates or DJIA increase) = 0.50 + 0.67 ? 0.45 = 0.72.

There are ten sprinters in the finals of a race. How many different ways can the gold, silver, and bronze medals be awarded? A. 120. B. 720. C. 1,440.

B Use calc permutation formula: 10P3 = 10! / (10-3)! = 720

For assets A and B we know the following: E(RA) = 0.10, E(RB) = 0.10, Var(RA) = 0.18, Var(RB) = 0.36 and the correlation of the returns is 0.6. What is the variance of the return of a portfolio that is equally invested in the two assets? A) 0.1500. B) 0.2114. C) 0.1102.

B You are not given the covariance in this problem but instead you are given the correlation coefficient and the variances of assets A and B from which you can determine the covariance by Covariance = (correlation of A, B) × Standard Deviation of A) × (Standard Deviation of B). Since it is an equally weighted portfolio, the solution is: [( 0.52 ) × 0.18 ] + [(0.52) × 0.36 ] + [ 2 × 0.5 × 0.5 × 0.6 × ( 0.180.5 ) × ( 0.360.5 )] = 0.045 + 0.09 + 0.0764 = 0.2114

The returns on assets C and D are strongly correlated with a correlation coefficient of 0.80. The variance of returns on C is 0.0009, and the variance of returns on D is 0.0036. What is the covariance of returns on C and D? A) 0.40110. B) 0.00144. C) 0.03020.

B r = Cov(C,D) / (σA x σB) σA = (0.0009)0.5 = 0.03 σB = (0.0036)0.5 = 0.06 0.8(0.03)(0.06) = 0.00144

A parking lot has 100 red and blue cars in it. 40% of the cars are red. 70% of the red cars have radios. 80% of the blue cars have radios. What is the probability that the car is red given that you already know that it has a radio? A) 47%. B) 37%. C) 28%.

B Given a set of prior probabilities for an event of interest, Bayes' formula is used to update the probability of the event, in this case that the car we already know has a radio is red. Bayes' formula says to divide the Probability of New Information given Event by the Unconditional Probability of New Information and multiply that result by the Prior Probability of the Event. In this case, P(red car has a radio) = 0.70 is divided by 0.76 (which is the Unconditional Probability of a car having a radio (40% are red of which 70% have radios) plus (60% are blue of which 80% have radios) or ((0.40) × (0.70)) + ((0.60) × (0.80)) = 0.76.) This result is then multiplied by the Prior Probability of a car being red, 0.40. The result is (0.70 / 0.76) × (0.40) = 0.37 or 37%.

Use the following data to calculate the standard deviation of the return: 50% chance of a 12% return 30% chance of a 10% return 20% chance of a 15% return A) 3.0%. B) 1.7%. C) 2.5%.

B The standard deviation is the positive square root of the variance. The variance is the expected value of the squared deviations around the expected value, weighted by the probability of each observation. The expected value is: (0.5) × (0.12) + (0.3) × (0.1) + (0.2) × (0.15) = 0.12. The variance is: (0.5) × (0.12 ? 0.12)2 + (0.3) × (0.1 ? 0.12)2 + (0.2) × (0.15 ? 0.12)2 = 0.0003. The standard deviation is the square root of 0.0003 = 0.017 or 1.7%. Use calc!!!!!

The unconditional probability of an event, given conditional probabilities, is determined by using the: A) addition rule of probability. B) total probability rule. C) multiplication rule of probability.

B The total probability rule is used to calculate the unconditional probability of an event from the conditional probabilities of the event given a mutually exclusive and exhaustive set of outcomes. The rule is expressed as: P(A) = P(A|B1)P(B1) + P(A|B2)P(B2) + ... + P(A|Bn)P(Bn)

Given P(X = 2) = 0.3, P(X = 3) = 0.4, P(X = 4) = 0.3. What is the variance of X? A) 3.0. B) 0.6. C) 0.3.

B The variance is the sum of the squared deviations from the expected value weighted by the probability of each outcome. The expected value is E(X) = 0.3 × 2 + 0.4 × 3 + 0.3 × 4 = 3. The variance is 0.3 × (2 ? 3)2 + 0.4 × (3 ? 3)2 + 0.3 × (4 ? 3)2 = 0.6. Use calc!!!

A discrete uniform distribution (each event has an equal probability of occurrence) has the following possible outcomes for X: [1, 2, 3, 4]. The variance of this distribution is closest to: A. 1.00. B. 1.25. C. 2.00.

B Expected value = (1/4)(1 + 2 + 3 + 4) = 2.5 Variance = (1/4)[(1 - 2.5)2 + (2 - 2.5)2 + (3- 2.5)2 + (4- 2.5)2] = 1.25 Note that since each observation is equally likely, each has 25% (1/4) chance of occurrence.

Two mutually exclusive events: A. always occur together. B. cannot occur together. C. can sometimes occur together.

B One or the other may occur, but not both.

Which of the following is an a priori probability? A) For a stock, based on prior patterns of up and down days, the probability of the stock having a down day tomorrow. B) The probability the Fed will lower interest rates prior to the end of the year. C) On a random draw, the probability of choosing a stock of a particular industry from the S& 500.

C A priori probability is based on formal reasoning and inspection. Given the number of stocks in the airline industry in the S&500 for example, the a priori probability of selecting an airline stock would be that number divided by 500.

Which of the following sets of numbers does NOT meet the requirements for a set of probabilities? A) (0.50, 0.50). B) (0.10, 0.20, 0.30, 0.40). C) (0.10, 0.20, 0.30, 0.40, 0.50).

C A set of probabilities must sum to one.

Each lottery ticket discloses the odds of winning. These odds are based on: A) past lottery history. B) the best estimate of the Department of Gaming. C) a priori probability.

C An a priori probability is based on formal reasoning rather than on historical results or subjective opinion.

An empirical probability is one that is: A) supported by formal reasoning. B) determined by mathematical principles. C) derived from analyzing past data.

C An empirical probability is one that is derived from analyzing past data. For example, a basketball player has scored at least 22 points in each of the season's 18 games. Therefore, there is a high probability that he will score 22 points in tonight's game.

Which of the following statements about the defining properties of probability is most accurate? A) The probability of any event is between 0 and 1, exclusive. B) The sum of the probabilities of events E1 though Ex equals one if the events are mutually exclusive or exhaustive. C) If the device that generates an event is not fair, the events can be mutually exclusive and exhaustive.

C Even if the device that generates an event is not fair, the events can be mutually exclusive and exhaustive. Consider a standard die with the possible outcomes of 1,2,3,4,5 and 6. The P(2 or 4 or 6) = 0.50 and P(1 or 3 or 5) = 0.50, and thus the probabilities sum to 1 and are mutually exclusive and exhaustive. An unfair die would not change this. Both remaining statements are false. The probability of any event is between 0 and 1, inclusive. It is possible that the probability of an event could equal 0 or 1, or any point in between. The sum of the probabilities of events E1 though Ex equals 1 if the events are mutually exclusive and exhaustive.

If two events are mutually exclusive, the probability that they both will occur at the same time is: A) 0.50. B) Cannot be determined from the information given. C) 0.00.

C If two events are mutually exclusive, it is not possible to occur at the same time. Therefore, the P(A∩B) = 0.

If event A and event B cannot occur simultaneously, then events A and B are said to be: A) statistically independent. B) collectively exhaustive. C) mutually exclusive.

C If two events cannot occur together, the events are mutually exclusive. A good example is a coin flip: heads AND tails cannot occur on the same flip.

At a charity fundraiser there have been a total of 342 raffle tickets already sold. If a person then purchases two tickets rather than one, how much more likely are they to win? A) 2.10. B) 0.50. C) 1.99.

C If you purchase one ticket, the probability of your ticket being drawn is 1/343 or 0.00292. If you purchase two tickets, your probability becomes 2/344 or 0.00581, so you are 0.00581 / 0.00292 = 1.99 times more likely to win.

Helen Pedersen has all her money invested in either of two mutual funds (A and B). She knows that there is a 40% probability that fund A will rise in price and a 60% chance that fund B will rise in price if fund A rises in price. What is the probability that both fund A and fund B will rise in price? A) 1.00. B) 0.40. C) 0.24.

C P(A) = 0.40, P(B|A) = 0.60. Therefore, P(A?B) = P(A)P(B|A) = 0.40(0.60) = 0.24

The correlation coefficient for a series of returns on two investments is equal to 0.80. Their covariance of returns is 0.06974 . Which of the following are possible variances for the returns on the two investments? A) 0.02 and 0.44. B) 0.08 and 0.37. C) 0.04 and 0.19.

C The correlation coefficient is: 0.06974 / [(Std Dev A)(Std Dev B)] = 0.8. (Std Dev A)(Std Dev B) = 0.08718. Since the standard deviation is equal to the square root of the variance, each pair of variances can be converted to standard deviations and multiplied to see if they equal 0.08718. √0.04 = 0.20 and √0.19 = 0.43589. The product of these equals 0.08718.

If given the standard deviations of the returns of two assets and the correlation between the two assets, which of the following would an analyst least likely be able to derive from these? A) Strength of the linear relationship between the two. B) Covariance between the returns. C) Expected returns.

C The correlations and standard deviations cannot give a measure of central tendency, such as the expected value.

There is an 80% chance that the economy will be good next year and a 20% chance that it will be bad. If the economy is good, there is a 60% chance that XYZ Incorporated will have EPS of $3.00 and a 40% chance that their earnings will be $2.50. If the economy is bad, there is a 70% chance that XYZ Incorporated will have EPS of $1.50 and a 30% chance that their earnings will be $1.00. What is the firm's expected EPS? A) $2.00. B) $4.16. C) $2.51.

C The expected EPS is calculated by multiplying the probability of the economic environment by the probability of the particular EPS and the EPS in each case. The expected EPS in all four outcomes are then summed to arrive at the expected EPS: (0.80 × 0.60 × $3.00) + (0.80 × 0.40 × $2.50) + (0.20 × 0.70 × $1.50) + (0.20 × 0.30 × $1.00) = $1.44 + $0.80 + $0.21 + $0.06 = $2.51.

For assets A and B we know the following: E(RA) = 0.10, E(RB) = 0.20, Var(RA) = 0.25, Var(RB) = 0.36 and the correlation of the returns is 0.6. What is the expected return of a portfolio that is equally invested in the two assets? A) 0.3050. B) 0.2275. C) 0.1500.

C The expected return of a portfolio composed of n-assets is the weighted average of the expected returns of the assets in the portfolio: ((w1) × (E(R1)) + ((w2) × (E(R2)) = (0.5 × 0.1) + (0.5 × 0.2) = 0.15.

The multiplication rule of probability is used to calculate the: A) probability of at least one of two events. B) unconditional probability of an event, given conditional probabilities. C) joint probability of two events.

C The multiplication rule of probability is stated as: P(AB) = P(A|B) × P(B), where P(AB) is the joint probability of events A and B.

Which of the following statements about counting methods is least accurate? A) The labeling formula determines the number of different ways to assign a given number of different labels to a set of objects. B) The multiplication rule of counting is used to determine the number of different ways to choose one object from each of two or more groups. C) The combination formula determines the number of different ways a group of objects can be drawn in a specific order from a larger sized group of objects.

C The permutation formula is used to find the number of possible ways to draw r objects from a set of n objects when the order in which the objects are drawn matters. The combination formula ("n choose r") is used to find the number of possible ways to draw r objects from a set of n objects when order is not important. The other statements are accurate.

Firm A can fall short, meet, or exceed its earnings forecast. Each of these events is equally likely. Whether firm A increases its dividend will depend upon these outcomes. Respectively, the probabilities of a dividend increase conditional on the firm falling short, meeting or exceeding the forecast are 20%, 30%, and 50%. The unconditional probability of a dividend increase is: A) 0.500. B) 1.000. C) 0.333.

C The unconditional probability is the weighted average of the conditional probabilities where the weights are the probabilities of the conditions. In this problem the three conditions fall short, meet, or exceed its earnings forecast are all equally likely. Therefore, the unconditional probability is the simple average of the three conditional probabilities: (0.2 + 0.3 + 0.5) ÷ 3.

In a given portfolio, half of the stocks have a beta greater than one. Of those with a beta greater than one, a third are in a computer-related business. What is the probability of a randomly drawn stock from the portfolio having both a beta greater than one and being in a computer-related business? A) 0.667. B) 0.333. C) 0.167.

C This is a joint probability. From the information: P(beta > 1) = 0.500 and P(comp. stock | beta > 1) = 0.333. Thus, the joint probability is the product of these two probabilities: (0.500) × (0.333) = 0.167.

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, but not both? A) 7.7%. B) 10.5%. C) 64.2%.

C Using the addition rule, the probability of being accepted at Harvard or Yale, but not both, is equal to: P(Harvard) + P(Yale) ? P(Harvard and Yale) = 0.25 + 0.42 ? 0.028 = 0.642 or 64.2%.

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) 0.125. B) 0.375. C) 0.625.

C 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

Jessica Fassler, options trader, recently wrote two put options on two different underlying stocks (AlphaDog Software and OmegaWolf Publishing), both with a strike price of $11.50. The probabilities that the prices of AlphaDog and OmegaWolf stock will decline below the strike price are 65% and 47%, respectively. The probability that at least one of the put options will fall below the strike price is approximately: A) 0.31. B) 1.00. C) 0.81.

C We calculate the probability that at least one of the options will fall below the strike price using the addition rule for probabilities (A represents AlphaDog, O represents OmegaWolf): P(A or O) = P(A) + P(O) ? P(A and O), where P(A and O) = P(A) × P(O) P(A or O) = 0.65 + 0.47 ? (0.65 × 0.47) = approximately 0.81

A parking lot has 100 red and blue cars in it. 40% of the cars are red. 70% of the red cars have radios. 80% of the blue cars have radios. What is the probability of selecting a car at random that is either red or has a radio? A) 76%. B) 28%. C) 88%.

C The addition rule for probabilities is used to determine the probability of at least one event among two or more events occurring, in this case a car being red or having a radio. To use the addition rule, the probabilities of each individual event are added together, and, if the events are not mutually exclusive, the joint probability of both events occurring at the same time is subtracted out: P(red or radio) = P(red) + P(radio) ? P(red and radio) = 0.40 + 0.76 ? 0.28 = 0.88 or 88%.

Which of the following values cannot be the probability of an event? A. 0.00. B. 1.00. C. 1.25.

C Probabilities may range from 0 (meaning no chance of occurrence) through 1 (which means a sure thing).

If events A and B are mutually exclusive, then: A. P(A| B) = P(A). B. P(AB) = P(A) x P(B). C. P(A or B) = P(A) + P(B).

C There is no intersection of events when events are mutually exclusive. P(A| B) = P(A) x P(B) is only true for independent events. Note that since A and B are mutually exclusive (cannot both happen), P(A| B) and P(AB) must both be equal to zero.

Two events are said to be independent if the occurrence of one event: A. means that the second event cannot occur. B. means that the second event is certain to occur. C. does not affect the probability of the occurrence of the other event.

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