An experiment is the act of making an observation or taking a measurement. For example, tossing dice and observing the numbers on top is an experiment.

A particular result of an experiment is called an outcome. It is the value assigned to a random variable. For example, there are six possible outcomes to the experiment of tossing dice.

A random variable is a quantity whose future outcomes are uncertain. For example, when you toss dice, the number on top is a random variable; you are unsure which number will come up.

Any outcome or specified set of outcomes of a random variable is called an event. For example, one event in the dice-tossing experiment is observing an odd number (three possible outcomes: 1, 3, 5).

Mutually exclusive events are events that only occurs one at a time. In the above example, event A (observe an odd number) and event B (observe an even number) are mutually exclusive because they cannot occur simultaneously; a number can be either odd or even, but cannot be both.

Exhaustive events are events that cover all the distinct possible outcomes. In the above example, event A and event B are exhaustive because they cover all six possible outcomes of the dice-tossing experiment.

As a general rule (and it is one you can use with confidence), if probabilities of two or more events add up to 1 and the events share no common outcomes at all, those events will always be mutually exclusive and exhaustive. However, it is important to remember that events can be mutually exclusive and exhaustive (as you've just seen), or can be one but not the other, or can be neither mutually exclusive nor exhaustive.

The two defining properties of probability

In general, probability is the likelihood that an event will happen. P(E) stands for "the probability of event E." The two defining properties of probability are:

• 0 ≤ P(E) ≤1: the probability of any event E is a number between 0 and 1. The probability of 0 means that the event can never happen and the probability of 1 means that the event is certain to happen.

• The sum of the probabilities of any list of mutually exclusive and exhaustive events equals 1. For example, the sum of the possibilities of two events (observing an odd number event and observing an even number event in the dice-tossing example) is 1, as these events are mutually exclusive and exhaustive.

Empirical probability

Empirical probability is a probability based on relative frequency of occurrence. It is estimated on the basis of historical data. For example, based on historical data over a 10-year period, the probability of default for real estate mortgage loans is 7%. We cannot estimate the empirical probability for an event without historical data. For empirical probabilities to be accurate, relationships must be stable over time.

Priori probability

Priori probability is a probability based on logical analysis rather than observation or personal judgment. For example, when you toss dice fairly, the probability of rolling an even number is 50%.

Empirical and priori probabilities generally do not vary from person to person, and they are often grouped as objective probabilities.

Subjective probability

Subjective probability is a probability based on personal or subjective judgment. For example, based on his own judgment, Bill believes that the probability that IBM's revenue will increase in 2005 is 60%.

According to the Dutch Book Theorem, one of the most important probability results theories for investments, inconsistent probabilities create profit opportunities. Investors should eliminate the profit opportunity and inconsistency through buy and sell decisions exploiting inconsistent probabilities.

Suppose that:

• If event E occurs, the values of two assets, A and B, will both rise.
• The price of asset A reflects a higher probability of event E than the price of asset B and thus inconsistent probabilities exist.

If all other factors are equal, asset A is overvalued compared with asset B.

• If event E does occur, the price of asset A will not rise as much as the price of asset B. This is because the occurrence of event E is mostly incorporated into the price of asset A.
• If event E does not occur, the prices of both assets will fall, but the price of asset A will decline more than the price of asset B. This is because, compared with asset B, the price of asset A understates the probability that event E may not occur.

Therefore, investors can profit by buying an undervalued asset (i.e., B) and selling an overvalued asset (i.e., A). Conservative investors will buy asset B and reduce or fully liquidate their position in asset A. Aggressive investors will buy asset B and short asset A. This strategy is known as the pairs arbitrage trade, which involves using the proceeds from the short sale of one stock to purchase another.

Note that the above discussion is based on the assumption that the occurrence of event E will increase the values of the two assets (A and B). If the occurrence of event E will reduce the value of assets A and B, asset B is overvalued if compared with asset A. To profit from inconsistent probabilities, investors should buy asset A and sell asset B.

Example

Suppose that if a hike in oil price occurs, the stock prices of American Airlines (AA) and British Airways (BA) will decline. The stock price of AA reflects a 0.7 probability of a hike in oil price, whereas the stock price of BA reflects only a 0.4 probability. In this situation, the stock of AA is undervalued when compared with the stock of BA. A conservative investor can profit by buying the stock of AA and reducing or eliminating his holdings in the stock of BA. An aggressive investor can profit by buying the stock of AA and shorting the stock of BA.

Practice Question 1

A probability experiment consists of picking one marble from a box that contains a red, a yellow, and a blue marble, and then flipping a coin once. Identify the sample space. (Let R = red, Y = yellow, B = blue, H = head, and T = tail.)

A. R, Y, B, H, T
B. RH, RT, YH, YT, BH, BT
C. RY, RB, YB, YR, BR, BY, HT, TH, HH, TT

Correct Answer: B

Practice Question 2

An experiment consists of removing one card from a deck of 52 cards. Let S = (H,D,C,P). Let event A be the event in which the card removed is a heart (H) or a diamond (D), A= (H,D). Let event B be the event in which the card removed is a spade (P) or a club (C), B = (C,P). Which of the following statements is incorrect?

A. A and B are disjoint events.
B. A and B are exhaustive events.
C. (D,C) is not a subset of S.

Correct Answer: C

Although (D,C) is not defined as an event, it is a subset of S.

Practice Question 3

Suppose that only three types of birds frequent your neighborhood and for a four-hour period, you record the birds you observe flying into your backyard. During that time, you observe 19 cardinals, 16 blue jays, and 12 robins. If each bird is equally likely to fly into your backyard, what is the probability that the next bird you observe is a robin?

Correct Answer: 0.255

p = 12 / (19 + 16 + 12) = 0.255

Practice Question 4

Suppose a traffic light cycles through green for 180 seconds, then yellow for 10 seconds, and then red for 110 seconds. Find the probability that at a certain time the light is green. Find the odds in favor of the light being green.

A. 0.01; 3:5
B. 0.60; 3:5
C. 0.60; 3:2

Correct Answer: C

p = 180 / (180 + 110 + 10) = 0.6
Odds: 180:(110 + 10) = 3:2

Practice Question 5

The ratio of males to females at OWJC is 3:5. P(female) = ______ (to nearest 0.1%)

A. 60%
B. 62.5%
C. 133.3%

Correct Answer: B

If the ratio of males to females is 3:5, then for every 8 students, 3 are male and 5 are female. Now, P(female) is ratio of success (female) to total. Thus, P(female) = 5:8 = 5/8 = 0.625 = 62.5%.

Practice Question 6

A card is randomly select from a standard 52-card deck. The odds in favor of a king are ______.

A. 4 : 48
B. 48 : 4
C. 4 : 52

Correct Answer: A

The odds in favor of an event is the ratio of success to failure. For a king, there are 4 chances of success and 48 chances of failure. So, odds in favor of a king are 4:48.

Practice Question 7

For the bar graph of random variable x shown below:

A. mean = 1.2
B. P(x > 0) = 0
C. P(x <= 5) = 1
D. expected value of x is 1

Correct Answer: C

The sum of all the P(x)s is 1 for any probability distribution. Now, P(x) <= 5 is the sum of all the P(x)s. So, P(x <= 5) = 1.

Practice Question 8

Which of the following is NOT a possible probability?

A. 25/100
B. 1.25
C. 1
D. 0

Correct Answer: B

Practice Question 9

You are given a discrete random variable, X, which has the following distribution: p(X=0, 1%, 2%, or 3%) = 25%, P(X > 3%) = 50%; P(X < 0) = 50%. Does this function satisfy the conditions for a probability function?

A. No
B. Yes
C. Not enough information to tell

Correct Answer: A

The two key requirements for a probability function are that 0 <= p(x) <= 1 and that the sum of the probabilities p(x) over all possible values of X is equal to 1. This function meets the first test but fails the second test. The second test fails because the sum of the probabilities equals 4 * 0.25 + 0.5 + 0.5 = 2.0

Practice Question 10

Which of the following cannot represent the probability of event A?

A. 1
B. 3/4
C. -1/2

Correct Answer: C

The correct answer is -1/2. This is not a legitimate probability since it is negative and 0 <= P(A) <= 1.

Practice Question 11

The probability that the unemployment rate will increase in January is 0.35. What is the probability that the unemployment rate will not increase in January?

A. 0.30
B. 0.35
C. 0.65

Correct Answer: C

The answer is 0.65 because the sum of the probabilities of any set of mutually exclusive and exhaustive events is 1. The probability that the unemployment rate will not increase in January is 1 - the probability that it will increase in January, or 1 - 0.35 = 0.65.

Practice Question 12

Classify each statement as an example of empirical probability, classical probability, or subjective probability.

I. The probability a randomly selected citizen will approve of the U.S. president is 0.75.
II. The probability a child will attend the same college as his father is 0.4.

Correct Answer: empirical, subjective

Practice Question 13

The probability of correctly answering a multiple-choice question with four possible answers by guessing is an example of which type of probability?

A. relative frequency or empirical
B. classical or priori
C. subjective

Correct Answer: B

The correct answer is classical or a priori since the probability can be determined by considering equally likely outcomes to the experiment.

Practice Question 14

You can enter a derivative contract that will pay $100 at the end of a year if the price of corn exceeds $3 per bushel or $50 if it is equal to $3 per bushel or lower. The probability that corn will exceed $3 by the end of one year is 50%. The current price of the contract is $60 and interest is 5% per year. What is the optimal strategy?

A. Invest $60 at 5% until the end of the year
B. Buy $3 per bushel worth of corn futures
C. Enter into the derivative contract for a cost of $60

Correct Answer: C

Enter into the derivative contract for a cost of $60; the expected payoff is 0.50 * $100 + 0.50 * $50 = $75. That is a 25% return on your investment in one year, greater than the 5% that could be made by investing the $60 at interest. This is an example of the investment consequences of inconsistent probabilities. The present value of the contract should be $75/1.05 = $71.43. Thus, an arbitrage opportunity is present. On an expected value basis, you can buy an asset worth $71.43 for only $60.

Practice Question 15

You are comparing two firms, Bex and Ajax Limited. The profitability of both firms stands to increase from reduction in income tax expenses. The price of Bex shares indicates a probability of 0.88 that income taxes will be reduced within the year. The price of Ajax Limited stock, however, reflects a 0.47 probability that income taxes will be reduced within that time frame. All other information related to valuation indicates that the two stocks appear comparably valued.

Which statement(s) is (are) TRUE?

I. The implied probabilities of 0.88 and 0.47 are inconsistent in that they create a potential profit opportunity.
II. The shares of Bex are relatively overvalued compared to Ajax, as their price incorporates a much higher probability of the favorable event (reduction in income taxes) than the shares of Ajax.
III. The shares of Ajax are relatively undervalued in comparison with the shares of Bex.

Correct Answer: I, II and III

Inconsistent probabilities arise when two or more assets are priced based upon different probabilities assigned to the same event. This creates opportunities for wise investors to take advantage of the inconsistencies.

A term that is often used for this is a pairs arbitrage trade. Pairs arbitrage trade is a trade in two closely related stocks involving the short-sale of one stock and the use of proceeds to purchase more of the other stock.

The theorem known as the Dutch Book Theorem states that inconsistent probabilities create opportunities for profit.

Consider the following scenario: Two stocks, A and B, in the same industry, in which an important announcement is pending, have share-prices which reflect the pending announcement differently. Suppose that an investor feels that stock A has the announcement factored into its price with a probability of 0.7, but stock B has the announcement factored into its price with a probability of just 0.4. Suppose also that the announcement is likely to be favorable for both stocks. One of two scenarios could exist:

1. Stock A is fairly valued and stock B is undervalued.
2. Stock A is overvalued and stock B is fairly valued.

In either case, stock B would seem to offer better value than stock A, and thus the investor, provided that he has confidence in his beliefs, could adopt the aggressive strategy of short-selling stock A and using the proceeds to purchase stock B.

Note that the costs of such a transaction would need to be taken into account in order to assess the viability of the trade. Ultimately, though, investors, through their buy and sell decisions, will eliminate the profit opportunity and inconsistency, as the share prices will alter by taking into account the demand for each stock.

Practice Question 16

An experiment consists of observing the total rate of return for the stocks in a portfolio for 2014. Let Y be the percentage total return rate for a stock. Let event A be the event that the total return rate for a stock in 2014 be greater than 1%. What is the sample space for this experiment?

A. S = {y|y>1}
B. S = {y|y>=1}
C. S = {y|y>=-1}

Correct Answer: C

The sample space gives us the possible outcomes for a stock's return rate. The return rate for a stock can be either positive or negative but cannot be less than -100%.

Practice Question 17

Which of the following is true regarding probability?

I. 0 < P(E) < 1: the probability of event E is a number between 0 and 1, exclusive.
II. The sum of the probabilities of any group of mutually exclusive events equals 1.
III. 0 <= P(E) <= 1: the probability of event E is a number between 0 and 1, inclusive.

A. III only
B. I and II
C. II and III

Correct Answer: A

Only III is true: 0 <= P(E) <= 1: the probability of event E is a number between 0 and 1, inclusive. A probability can equal 1 or 0; thus, statement I is false. Statement II would only be true if it read: "The sum of the probabilities of any group of mutually exclusive and exhaustive events equals 1." Throwing a 6-sided die can lead to 6 different outcomes, each with probability of 1/6. However, if the group of mutually exclusive outcomes only included results 1 and 2, the sum of those probabilities would not be 1. Such a group would not be exhaustive because it excludes the outcomes 3, 4, 5, 6.

Practice Question 18

Which of the following is correct about a probability distribution?

I. The sum of the probabilities of all possible outcomes must equal 1.
II. The probability of each outcome must be between 0 and 1 inclusive.
III. Outcomes must be mutually exclusive.

A. I and II
B. I and III
C. I, II and III

Correct Answer: C

All these statements describe properties of a probability distribution.

Practice Question 19

Out of the last 154 times that interest rates have been cut, the stock price of Company XYZ dropped 72 times. The probability that company XYZ's stock drops at the next rate cut is an example of what type of probability?

A. relative frequency or empirical
B. classical or a priori
C. subjective

Correct Answer: A

The correct answer is relative frequency or empirical since the probability of an event happening in the future is found by determining what fraction of the time the event occurred in the past.

Practice Question 20

An economic analyst says that the probability of Alan Greenspan announcing a rate cut next week is 0.45 or 45%. This is an example of what type of probability?

A. relative frequency or empirical
B. classical or a priori
C. subjective

Correct Answer: C

The correct answer is subjective since this probability is an educated guess based on the analyst's personal belief or experience.

Practice Question 21

A random variable (is) ______.

A. a quantity whose outcome is uncertain
B. covers all possible outcomes
C. any outcome or specified set of outcomes

Correct Answer: A

A random variable is a quantity whose outcome is uncertain.

Practice Question 22

Empirical probability is ______.

A. a technique where past information is used to determine future probabilities
B. a technique where current information is used to determine future probabilities
C. a form of probability that takes into account the fact that some other event has occurred which is likely to affect the chance of our event occurring

Correct Answer: A

Empirical probability is a technique where past information is used to determine future probabilities.

Practice Question 23

If an analyst estimates the probability of an event for which there is no historical record, this probability is best described as ______.

A. empirical
B. subjective
C. a priori

Correct Answer: B

An empirical probability cannot be calculated for an event not in the historical record. In this case, the analyst can make a personal assessment of the probability of the event without reference to any particular data. This is a subjective probability.