Probability concepts

Question 1

Random variable

Answer 1

quantity with uncertain outcomes

Question 2

Outcome

Answer 2

possible values

Question 3

event

Answer 3

specified set of outcomes

Question 4

Mutually exclusive events

Answer 4

Only one event can occur at a time

Question 5

Exhaustive events

Answer 5

Events cover all possible outcomes

Question 6

Probability

Answer 6

number between 0 and 1 that measures chance event will occur.

Question 7

empirical vs. subjective vs. a priori probabilities

Answer 7

Empirical - probability estimate using relative frequency based on historical data

Subjective - personal judgment of probability

A priori - probability based on logical analysis

Question 8

state probability as odds for and against

Answer 8

for = a/a+b
against = b/a+b

Question 9

unconditional vs. conditional probabilities

Answer 9

Unconditional - what is probability of event X

Conditional - what is probability of A given that B occurred
P(A | B) is “probability of A given B”
P(A | B) = P(AB)/P(B)

Question 10

multiplication, addition, and total probability rules

Answer 10

Multiplication: joint probability of two events

Addition: probability at least one of two events occur

Total: unconditional probability of event in terms of probabilities conditional on scenarios

Question 11

calculate and interpret joint probability of two events

Answer 11

P(AB) = P(A | B) * P(B)

Question 12

calculate and interpret probability that at least one of two events will occur

Answer 12

P(A or B) = P(A) + P(B) - P(AB)

Question 13

calculate and interpret joint probability of independent events

Answer 13

P(AB) = P(A)P(B)

Question 14

dependent vs. independent evnets

Answer 14

Occurrence of A does not depend on B and vice-a-versa.

Question 15

calculate and interpret unconditional probability using total probability rule

Answer 15

P(A) = P(A | S1)P(S1) + P(A | S2)P(S2) + … P (A|Sn)P(Sn)

Question 16

explain use of conditional expectation in investment

Answer 16

Add up possible expected values given mutually exclusive and exhaustive scenarios. Update with new info.

Question 17

explain tree diagram

Answer 17

Nodes are different branches. Each decision point has 1 total probability. Can add up expected value based on weighted probabilities of each outcome.

Question 18

calculate and interpret covariance and correlation

Answer 18

Cov(Ri,Rj) = E [(Ri - ERi)(Rj - ERj)]

Make Covariance matrix.
Cov(R1,R2,R3) = w1^2 * (varR1) + w2^2 * (varR2) + w3^2 * (varR3) + 2w1w2cov(R1,R2) + 2w1w3cov(R1,R3) + 2w2w3ccov(R2,R3)

Corr(Ri,Rj) = Cov(Ri,Rj) / σ(Ri) * σ(Rj) 
σ = square root of covariance

0 - uncorrelated
Positive - positive linear relationship
Negative - inverse linear relationship

Question 19

calculate and interpret expected value, variance, standard deviation

Answer 19

E(X) = Σ P(Xi)(Xi)

σ^2 = Σ P(Xi) [Xi - E(X)]^2

Std deviation: square root of variance (σ)

Question 20

calculate and interpret covariance given joint probability

Answer 20

Make table of joint probabilities and multiply each joint probability by the respective expected value. Result is the expected return of each asset.

Question 21

calculate and interpret updated probability using Bayes’ formula

Answer 21

In light of new info, what is updated probability of event?

P (Event | Info) = P(Info | Event) / P(Info) * P(Event)

Write out conditional probabilities for Info given all Events.
P(Info) = Σ P(Info | Event) * P(Event)

Question 22

identify best method to solve counting problem

Answer 22

If infinite outcomes, no tool.

If assign each member to one slot, use factorial.

If want to count r objects from n and order doesn’t matter, use combination formula

If want to count number of ways choose r objects from n and order matters, use permutation formula

Question 23

Solve counting problems using factorial, combination, and permutation concepts

Answer 23

Factorial: n! (on calc)

Combination: nCr = n! / (n-r)! r!

Permutation: nPr = n! / (n-r)!