CFA Level 1 - Quantitative Methods Session 2 - Reading 8

CFA Level 1 - Quantitative Methods Session 2 - Reading 8 (Notes, Practice Questions, Sample Questions) 1. 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% [Explanation: (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%] 2. What is the probability of a bull market next year? A)20%. B)50%. C)32% [Explanation: (C) 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%] 3. For a stock, which of the following is least likely a random variable? Its: A)current ratio. B)most recent closing price. C)stock symbol [Explanation: (C) 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] 4. If two events are mutually exclusive, the probability that they both will occur at the same time is: A)0.50. B)0.00. C)Cannot be determined from the information given [Explanation: (B) If two events are mutually exclusive, it is not possible to occur at the same time. Therefore, the P(A∩B) = 0] 5. Which of the following statements about probability is most accurate? A)An outcome is the calculated probability of an event. B)A conditional probability is the probability that two or more events will happen concurrently. C)An event is a set of one or more possible values of a random variable [Explanation: (C) 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] 6. If event A and event B cannot occur simultaneously, then events A and B are said to be: A)mutually exclusive. B)statistically independent. C)collectively exhaustive [Explanation: (A) 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] 7. 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% [Explanation: (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%] 8. Which of the following is an a priori probability? A)On a random draw, the probability of choosing a stock of a particular industry from the S&P 500. B)For a stock, based on prior patterns of up and down days, the probability of the stock having a down day tomorrow. C)The probability the Fed will lower interest rates prior to the end of the year [Explanation: (A) A priori probability is based on formal reasoning and inspection. Given the number of stocks in the airline industry in the S&P500 for example, the a priori probability of selecting an airline stock would be that number divided by 500] 9. Each lottery ticket discloses the odds of winning. These odds are based on: A)a priori probability. B)past lottery history. C)the best estimate of the Department of Gaming [Explanation: (A) An a priori probability is based on formal reasoning rather than on historical results or subjective opinion] 10. 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)20%. B)1 to 5. C)1 to 4 [Explanation: (C) 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] 11. 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 [Explanation: (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] 12. An empirical probability is one that is: A)derived from analyzing past data. B)supported by formal reasoning. C)determined by mathematical principles [Explanation: (A) 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] 13. Which of the following sets of numbers does NOT meet the requirements for a set of probabilities? A)(0.10, 0.20, 0.30, 0.40, 0.50). B)(0.50, 0.50). C)(0.10, 0.20, 0.30, 0.40) [Explanation: (A) A set of probabilities must sum to one] 14. Which of the following is an empirical probability? A)The probability the Fed will lower interest rates prior to the end of the year. B)On a random draw, the probability of choosing a stock of a particular industry from the S&P 500 based on the number of firms. C)For a stock, based on prior patterns of up and down days, the probability of the stock having a down day tomorrow [Explanation: (C) There are three types of probabilities: a priori, empirical, and subjective. An empirical probability is calculated by analyzing past data] 15. If the probability of an event is 0.10, what are the odds for the event occurring? A)One to ten. B)Nine to one. C)One to nine [Explanation: (C) 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] 16. 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 [Explanation: (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] 17. 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)0.92. B)0.76. C)0.06. [Explanation: (A) 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%] 18. If the probability of an event is 0.20, what are the odds against the event occurring? A)Four to one. B)Five to one. C)One to four [Explanation: (A) 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] 19. If the odds against an event occurring are twelve to one, what is the probability that it will occur? A)0.0833. B)0.9231. C)0.0769 [Explanation: (C) 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] 20. Let A and B be two mutually exclusive events with P(A) = 0.40 and P(B) = 0.20. Therefore: A)P(B|A) = 0.20. B)P(A and B) = 0. C)P(A and B) = 0.08 [Explanation: (B) If the two evens are mutually exclusive, the probability of both ocurring is zero] 21. 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 [Explanation: (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] 22. The unconditional probability of an event, given conditional probabilities, is determined by using the: A)addition rule of probability. B)multiplication rule of probability. C)total probability rule [Explanation: (C) The total probability rule us 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)] 23. The multiplication rule of probability is used to calculate the: A)joint probability of two events. B)probability of at least one of two events. C)unconditional probability of an event, given conditional probabilities [Explanation: (A) 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] 24. 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)0.24. B)0.50. C)0.86 [Explanation: (C) 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] 25. 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%. [Explanation: (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% Radio No Radio Red 28 12 40 Blue 48 12 60 76 24 100] 26. What is the probability of selecting a car at random that is either red or has a radio? A)76%. B)88%. C)28% [Explanation: (B) 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%] 27. What is the probability that the car is red given that you already know that it has a radio? A)47%. B)28%. C)37% [Explanation: (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 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%] 28. 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)60%. B)95%. C)15%. [Explanation: (A) 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%] 29. 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 [Explanation: (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] 30. 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)0.80. B)0.60. C)0.70. [Explanation: (A) P(AB) = P(A|B) × P(B)0.68 / 0.85 = 0.80] 31. 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)0.675. B)0.306. C)0.576 [Explanation: (C) 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] 32. Pat Binder, CFA, is examining the effect of an inverted yield curve on the stock market. She determines that in the past century, 75% of the times the yield curve has inverted, a bear market in stocks began within the next 12 months. Binder believes the probability of an inverted yield curve in the next year is 20%. Binder's estimate of the probability that there will be an inverted yield curve in the next year followed by a bear market is closest to: A)50%. B)15%. C)38%. [Explanation: (B) This is a joint probability. From the information: P(Bear Market given inverted yield curve) = 0.75 and P(inverted yield curve) = 0.20. The joint probability is the product of these two probabilities: (0.75)(0.20) = 0.15] 33. 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)64.2%. C)10.5%. [Explanation: (B) 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%] 34. There is a 30% probability of rain this afternoon. There is a 10% probability of having an umbrella if it rains. What is the chance of it raining and having an umbrella? A)40%. B)33%. C)3% [Explanation: (C) P(A) = 0.30. P(B | A) = 0.10. P(AB) = (0.30)(0.10) = 0.03 or 3%] 35. If two events are independent, the probability that they both will occur is: A)0.50. B)Cannot be determined from the information given. C)0.00 [Explanation: (B) If two events are independent, their probability of their joint occurrence is computed as follows: P(A∩B) = P(A) × P(B). Since we are not given any information on the respective probabilities of A or B, there is not enough information] 36. A very large company has twice as many male employees relative to female employees. If a random sample of four employees is selected, what is the probability that all four employees selected are female? A)0.3333. B)0.0625. C)0.0123 [Explanation: (C) Since there are twice as many male employees to female employees, P(male) = 2/3 and P(female) = 1/3. Therefore, the probability of 4 “successes” = (0.333)4 = 0.0123] 37. 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)0.95. C)0.72 [Explanation: (C) 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] 38. 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. [Explanation: (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] 39. An analyst has a list of 20 bonds of which 14 are callable, and five have warrants attached to them. Two of the callable bonds have warrants attached to them. If a single bond is chosen at random, what is the probability of choosing a callable bond or a bond with a warrant? A)0.70. B)0.85. C)0.55 [Explanation: (B) This requires the addition formula, P(callable) + P(warrants) – P(callable and warrants) = P(callable or warrants) = 14/20 + 5/20 – 2/20 = 17/20 = 0.85] 40. For the task of arranging a given number of items without any sub-groups, this would require: A)the labeling formula. B)the permutation formula. C)only the factorial function [Explanation: (C) The factorial function, denoted n!, tells how many different ways n items can be arranged where all the items are included] 41. 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 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 multiplication rule of counting is used to determine the number of different ways to choose one object from each of two or more groups [Explanation: (B) 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] 42. 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)120. B)25. C)252. [Explanation: (C) 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] 43. 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 [Explanation: (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]