(8) Quantitative Methods: Probability Concepts (kyle created) Flashcards
Random Variable
A quantity whose outcomes are unknown
Outcome
Possible values
Event
A specified set of outcomes (point or range); denoted by capital letters
The two defining properties of a probability are as follows:
= P (A)
- P (A) is less than or equal to 1 or greater than or equal to zero
- The sum of all probabilities sums up to 1
Mutually exclusive events
If one event happens, another event can’t
Exhaustive events
Covers all possible outcomes
The two elements needed to solve a probability problem are:
- Set of all distinct possible outcomes
2. The probability distribution
Empirical Probability
- Based on historical observations
- Past is assumed to be representative of the future
- Historical period must include occurrences of the event
Subjective probability
- Adjust an empirical probability based on intuition or experience
- This occurs when there is a complete lack of empirical observations or
- to make a personal assessment
A priori probability
arrived at based on deductive reasoning
Odds for E. What is the formula and meaning?
P (E) / 1 - P(E);
Ex: P(E) = 10%
Odds for E = .1 / (1 - .10)
Odds for E = .1/.9 or 1 to 9
This is saying for each occurrence of event E, we should expect 9 events of non-occurrence
If Odds for E = 1 to 9 then to get the probability you take 1 / (1+9)
Odds against E. What is the formula and meaning?
1 - P (E) / P(E);
Ex: P(E) = 10%
Odds against E = .9 / .1 or 9 to 1
This is saying for every 9 non-occurrence of event E, we should expect 1 occurrence of the event
If Odds against E = 9 to 1 then to get the probability you take 1 / (9+1)
Probability: Terminology (5)
- Random Variable: Uncertain number
- Outcome: Realization of random variables
- Event: Set of one or more outcomes
- Mutually exclusive: cannot both happen
5: Exhaustive: Set of events includes all possible outcomes
Probability: Types (3)
- Empirical: Based on analysis of data
- Subjective: Based on personal perception
- A priori: Based on reasoning, not experience
Probability: Odds for or Against (2)
Odd happening/ odds not happening.
Given probability that a horse will win a race= 20%
Odds for: .20/ (1-.2)= .2/.8= 1/4 or 1 to 4 Odds against: (1-.2) .2= .8/.2= 4 or 4 to 1
Probability: Conditional vs Unconditional (2)
Two types of probability:
- Unconditional P (A), the probability of an event regardless of the outcomes of other events
- Conditional P (A|B), the probability of A given that B has occurred (e.g. The probability that Marley will be up, given the red raises interests rates)
Joint Probability (Multiplication rule of probability)
Notation is P (AB) which means the probability of A and B
The probability that both events will occur is their joint probability
Examples using conditional probability:
P (interest rates will increase) = P (A) = 40%
P (recession given a rate increase) = P (B|A) = 70%
Probability of a recession and an increase in rates,
P (BA) =P(B|A) x P(A) =.7 x .4 = 28%
Addition Rule for Probabilities
- P (A or B) = P(A) + P(B) - P(AB)
- We must subtract the joint probability P (AB)
- At least one of two events will occur
Formula for the probability of A given B; P (A|B) =
P(AB) / P(B)
Multiplication rule for independent events
When two events are independent of each other, the joint probability of A and B = P(A) x P(B)
Total probability rule formula is
P(A) = P(A|S1)xP(S1) + P(A|S2)xP(S2)….+ P(A|Sn)xP(Sn)
A is the event
S1 is scenario 1
S2 is scenario 2
Portfolio expected return
Weighted average of the expected returns on the different securities
Covariance of returns is negative if these two conditions are met
- The return on one asset is above its expected value
- The return on the other asset tends to be below its expected value
This is an inverse relationship
Covariance of returns is equal to 0 when
The return on the assets are unrelated
