probability concepts (Reading 8)  

Learning Outcome Statements (LOS)

a

Define a random variable, an outcome, an event, mutually exclusive events, and exhaustive events:

     A random variable is an uncertain value determined by chance.

     An outcome is the realization of a random variable.

     An event is a set of one or more outcomes. Two events that cannot both occur are termed “mutually exclusive” and a set of events that includes all possible outcomes is an “exhaustive” set of events.

b

State the two defining properties of probability and distinguish among empirical, subjective, and a priori probabilities:

     The two properties of probability are:

·         The sum of the probabilities of all possible mutually exclusive events is 1.

·         The probability of any event cannot be greater than 1 or less than 0.

     A priori probability measures predetermined probabilities based on well-defined inputs; empirical probability is an informed guess.

c     

State the probability of an event in terms of odds for and against the event:

     Probabilities can be stated as odds that an event will or will not occur. If the probability of an event is A out of B trials (A/B), the “odds against” are (B-A) to A.

d    

Distinguish between unconditional and conditional probabilities:

     Unconditional probability (marginal probability) is the probability of an event occurring.

     Conditional probability, P (A|B), is the probability of an event A occurring given that event B has occurred.

e  

Explain the multiplication, addition, and total probability rules:

     The multiplication rule of probability is used to determine the joint probability of two events:

     The addition rule of probability is used to determine the probability the probability that at least one of two events will occur:

     The total probability rule is used to determine the unconditional probability of an event, given conditional probability:

     Where B₁, B₂…B_N is a mutually exclusive and exhaustive set of outcomes.       

f 

Calculate and interpret 1) the joint probability of two events, 2) the probability that at least one of two events will occur, given the probability of each and the joint probability of the two events, and 3) a joint probability of any number of independent events:

     The joint probability of two events, P (AB), is the probability that they will both occur. . For independent events,

     The probability that at least one of two events will occur is. For mutually exclusive events,, since P(AB)=0.

     The joint probability of any number of independent events is the product of their individual probabilities.

g

Distinguish between dependent and independent events:

     The probability of an independent event is unaffected by the occurrence of other events, but the probability of a dependent event is changed by the occurrence of another event. Events A and B are independent if and only if:

, or equivalently,

h

Calculate and interpret an unconditional probability using the total probability rule:

     Using the total probability rule, the unconditional probability of A is the probability weighted sum of the conditional probabilities:

,

     Where B_i is a set of mutually exclusive and exhaustive events.

i  

Explain the use of conditional expectation in investment applications:

     Conditional expected values depend on the outcome of some other event.

     Forecasts of expected values for a stock’s return, earnings, and dividends can be refined; using conditional expected values, when new information arrives that affects the expected outcome.

j    

Explain the use of a tree diagram to represent an investment problem:

     A tree diagram shows the probabilities of two events and the conditional probabilities of two subsequent events.

k

Calculate and interpret covariance and correlation:

     Covariance measures the extent to which two random variables tend to be above and below their respective means for each joint realization. It can be calculated as:

     Correlation is a standardized measure of association between two random variables; it ranges in value from -1 to +1 and is equal to.

l  

Calculate and interpret the expected value, variance, and standard deviation of a random variable and of returns on a portfolio:

      The expected value of a random variable, E(X), equals.

      The variance of a random variable, Var(X), equals

      Standard deviation:

      The expected return and variance of a 2-asset portfolio are given by:

m

Calculate and interpret covariance given a joint probability function:

     Given the joint probabilities for X_i and Y_i, i.e., P(X_iY_i), the covariance is calculated as:

n

Calculate and interpret an updated probability using Bayes’ formula:

     Bayes’ formula for updating probabilities based on the occurrence of an event O is:

o

Identify the most appropriate method to solve a particular counting problem, and solve counting problems using the factorial, combination, and permutation notations.

     The number of ways to order n objects is n factorial,. There are ways to assign k different labels to n items, where n_i is the number of items with the label i.

     The number of ways to choose a subset of size R from a set of size n when order doesn’t matter is combinations; when order matters, there arepermutations.

Formulas:

Exercise Problems:

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

A.    Empirical probability

B.     Subjective probability

C.     A priori probability

Ans: B; in some case, we may adjust an empirical probability to account for perceptions of changing relationships. In other case, we have no empirical probability to use as all. We may also make a personal assessment of probability without reference of any particular data. Each of these three types of probability is a subjective probability.

A is incorrect; in investment, we often estimate the probability of an event as a relative frequency of occurrence based on historical data. This method produces an empirical probability.

C is incorrect; in a more narrow range of well-defined problems, we can sometimes deduce probability by reasoning about the problem. The resulting probability is an a priori probability。  

2.    The probability of event A is 40%. The probability of event B is 65%. The joint probability of AB is 25%. The probability that A or B occurs or both occur is closest to:

A.    40%

B.     65%

C.     80%

Ans: C; The addition rule of probability is used to determine the probability the probability that at least one of two events will occur:

3.    A fundamental analyst studying 50 potential companies for inclusion in his stock portfolio uses the following two screening criteria:

Assuming that the screening criteria are independent, the probability that a given company will meet all three screening criteria is closest to:

A.    6%

B.     20%

C.     50%

Ans: A; The joint probability of two independent events, 

4.    Which of the following statements best describes the relationship between correlation and covariance? The correlation between two random variables is their covariance standardized by the product of the variables’:

A.    Variances

B.     Standard deviations

C.     Coefficients of variation

Ans: B; Correlation is a standardized measure of association between two random variables; it ranges in value from -1 to +1 and is equal to.

5.    The joint probability of returns for securities A and B are as follows:

The correlation between securities A and B is closest to:

A.    0.0428

B.     0.0001

C.     0.0023

Ans: A;

6.    How many permutations are possible when choose 4 objects from a total of 10 objects?

A.    30

B.     210

C.     5,040

Ans: C; The number of ways to choose a subset of size R from a set of size n when order matters, there arepermutations.

So in this problem,