CFA. Level I. R9 - Probability Concepts

Mutually exclusive events

Events that cannot occur at the same time.

Five Guidelines for Labeling

- The multiplication rule of counting is used when there are two or more groups. The key is that only one item may be selected from each group. If there are k steps required to complete a task and each step can be done in n ways, the number of different ways to complete the task is n1 × n2 × ... × nk. - Factorial is used by itself when there are no groups—we are only arranging a given set of n items. Given n items, there are n! ways of arranging them. - The labeling formula applies to three or more subgroups of predetermined size. Each element of the entire group must be assigned a place, or label, in one of the three or more subgroups. - The combination formula applies to only two groups of predetermined size. Look for the word "choose" or "combination." - The permutation formula applies to only two groups of predetermined size. Look for a specific reference to "order" being important.

Tree diagram

A diagram used to show the total number of possible outcomes.

Permutation formula

A permutation is a specific ordering of a group of objects. The question of how many different groups of size r in specific order can be chosen from n objects is answered by the permutation formula. The permutation formula implies that there are r! more ways to choose r items if the order of selection is important than if order is not important. That is, nPr = r! × nCr .

Combination formula

A special case of labeling arises when the number of labels equals 2 (k = 2). That is, the n items can only be in one of two groups, and n1 + n2 = n. In this case, we can let r = n1 and n2 = n − r. The combination formula applies to only two groups of predetermined size. Look for the word "choose" or "combination."

Correlation coefficient

A statistical measure of the extent to which two factors vary together, and thus of how well either factor predicts the other. Properties of correlation of two random variables Ri and Rj are summarized here: - Correlation measures the strength of the linear relationship between two random variables. - Correlation has no units. - The correlation ranges from -1 to +1. That is, -1 ≤ Corr(Ri, Rj) ≤ +1. - If Corr(Ri, Rj) = 1.0, the random variables have perfect positive correlation. This means that a movement in one random variable results in a proportional positive movement in the other relative to its mean. - If Corr(Ri, Rj) = -1.0, the random variables have perfect negative correlation. - If Corr(Ri, Rj) = 0, there is no linear relationship between the variables, indicating that prediction of Ri cannot be made on the basis of Rj using linear methods.

Odds

That an event will or will not occur is an alternative way of expressing probabilities.

Expected value

The expected value of a random variable is the weighted average of the possible outcomes for the variable. _____ can be calculated using conditional probabilities. An analyst would use a conditional expected value to revise his expectations when new information arrives.

Exhaustive events

_____ are those that include all possible outcomes.

Portfolio Expected Value

_____ can be determined using the properties of the individual assets in the portfolio. To do this, it is necessary to establish the portfolio weight for each asset.

Objective probabilities

_____ derived mathematically from reliable historical data such as empirical and a priori probabilities

Empirical probability

_____ involves conducting an experiment to observe the frequency with which an event occurs. _____ is established by analyzing past data.

Covariance

_____ is a measure of how two assets move together. It is the expected value of the product of the deviations of the two random variables from their respective expected values. The following are properties of the covariance: - The covariance is a general representation of the same concept as the variance. That is, the variance measures how a random variable moves with itself, and the covariance measures how one random variable moves with another random variable. - The covariance of RA with itself is equal to the variance of RA; that is, Cov(RA,RA) = Var(RA). - The covariance may range from negative infinity to positive infinity.

An event

_____ is a single outcome or a set of outcomes.

An outcome

_____ is an observed value of a random variable.

Random variable

_____ is an uncertain quantity/number.

An a priori probability

_____ is determined using a formal reasoning and inspection process.

Portfolio Standard Deviation

_____ is not just a weighted average of the standard deviation of the component securities' risk.

A conditional probability

_____ is one where the occurrence of one event affects the probability of the occurrence of another event. The key word to watch for here is "given." Using probability notation, "the probability of A given the occurrence of B" is expressed as P(A | B), where the vertical bar (|) indicates "given," or "conditional upon." A conditional probability of an occurrence is also called its likelihood.

Subjective probability

_____ is the least formal method of developing probabilities and involves the use of personal judgment. An analyst may know many things about a firm's performance and have expectations about the overall market that are all used to arrive at a subjective probability

Multiplication rule of probability

_____ is used to determine the joint probability of two events. - A and B indicate multiplication rule

Addition rule of probability

_____ is used to determine the probability that at least one of two events will occur. - A or B indicate multiplication rule

Bayes' formula

_____ is used to update a given set of prior probabilities for a given event in response to the arrival of new information.

Independent events

_____ refer to events for which the occurrence of one has no influence on the occurrence of the others. The definition of independent events can be expressed in terms of conditional probabilities. Events A and B are independent if and only if: P(A | B) = P(A), or P(B | A) = P(B). A die has "no memory." Therefore, the event of rolling a 4 on the second toss is independent of rolling a 4 on the first toss. This idea may be expressed as: P(4 on second toss | 4 on first toss) = P(4 on second toss) = 1/6 or 0.167

Unconditional probability

_____ refers to the probability of an event regardless of the past or future occurrence of other events. If we are concerned with the probability of an economic recession, regardless of the occurrence of changes in interest rates or inflation, we are concerned with the unconditional probability of a recession.

Labeling

_____ refers to the situation where there are n items that can each receive one of k different labels.

Total probability rule

_____ used to determine the unconditional probability of an event, given conditional probabilities