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# Probability Concepts

## 1. Probability Concepts

By Dias Kulzhanov## 2. Definition of Probability

Probability is a number between 0 and 1 that describes the chance

that a stated event will occur.

An event is a specified set of outcomes of a random variable.

Mutually exclusive events can occur only one at a time. Exhaustive

events cover or contain all possible outcomes.

The two defining properties of a probability are, first, that 0 ≤ P(E) ≤ 1

where P(E) denotes the probability of an event E), and second, that

the sum of the probabilities of any set of mutually exclusive and

exhaustive events equals 1.

A probability estimated from data as a relative frequency of

occurrence is an empirical probability. A probability drawing on

personal or subjective judgment is a subjective probability. A

probability obtained based on logical analysis is an a priori

probability.

## 3. Probability Stated as Odds

• Odds for E = P(E)/[1 − P(E)]. The odds for E are the probability of Edivided by 1 minus the probability of E. Given odds for E of “a to b,”

the implied probability of E is a/(a + b).

In the example, the statement that the odds for the company’s EPS

for FY2014 beating $0.69 are 1 to 7 means that the speaker believes

the probability of the event is 1/(1 + 7) = 1/8 = 0.125.

• Odds against E = [1 − P(E)]/P(E), the reciprocal of odds for E. Given

odds against E of “a to b,” the implied probability of E is b/(a+ b).

The statement that the odds against the company’s EPS for FY2014

beating $0.69 are 15 to 1 is consistent with a belief that the

probability of the event is 1/(1 + 15) = 1/16 = 0.0625.

## 4. Unconditional and conditional probability

• The probability in answer to the straightforward question “What isthe probability of this event A?” is an unconditional probability,

denoted P(A). Unconditional probability is also frequently referred to

as marginal probability.

• Contrast the question “What is the probability of A?” with the

question “What is the probability of A, given that B has occurred?”

The probability in answer to this last question is a conditional

probability, denoted P(A | B) (read: “the probability of A given B”).

## 5. Multiplication and Additional Rules of probability

## 6. The Total Probability Rule

## 7. Expected value

Variance of expected valueConditional expected values

## 8. Portfolio Expected Return

## 9.

Portfolio variance of return## 10.

## 11. Covariance and Correlation

• Covariance is a measure of the co-movement between randomvariables

• The calculation of covariance in a forward-looking sense requires

the specification of a joint probability function, which gives the

probability of joint occurrences of values of the two random

variables.

• Correlation is a number between −1 and +1 that measures the comovement (linear association) between two random variables.