Probability Concepts Flashcards
Empirical probability
- based on past data
- analyzing the frequence of an event’s occurrence in the past
Priori probability
- based on reasoning / logic
Subjective probability
- informed guess based on personal judgement
Odds for an event occurring:
- Odds for E = P(E) / [1 - P(E)]
Implied probability of E given “a to b”
- implied probability of E = a / (a + b)
Odds against an event occurring:
- odds against E = [1 - P(E)] / P(E)
Implied probability against E given “a to b”
- implied probability against E is = b / (a + b)
Unconditional probability, aka…
the ‘marginal’ probability
Unconditional probability def
- the probability of an event occurring irrespective of the occurrence of other events
- P(A)
Conditional probability
- the probability of an event occurring given that another event has occurred
- P(A|B)
Multiplication rule is used for:
- used for determining the joint probability of two events
- P(AB)
Multiplication rule formula:
P(AB) = P(A|B) * P(B)
also
P(BA) = P(B|A) * P(A)
The probability that both A AND B will occure
Formula for computing conditional probabilities:
- P(A|B) = P(AB) / P(B)
Addition Rule for probability:
- used to determine the probability that at least one of the events will occur
- the probability that A OR B occurs, OR both occur
Addition Rule probability formula
and formula if mutually exclusive
- P(A or B) = P(A) + P(B) - P(AB)
if mutually exclusive:
- P (A or B) = P(A) + P(B)
Multiplication rule for independent events =
= P(AB) = P(A) * P(B)
Addition rule for independent events =
- P(A or B) = P(A) + P(B) - P(AB)
* the addition rule does not change
The Total Probability Rule …
- TPR enables us to state unconditional probabilities in terms of conditional probabilities
S= scenario Sc = non-S
Total Probability Rule formulas for P(A) =
- =P(A) = P(AS) + P(ASc)
- = P(A) = P(A|S)P(S) + P(A|Sc)P(Sc)
Expected Value of a Random Variable is
- the probability-weighted average of the possible outcomes of the random variable
State of Econ. | Prob. | CF. | E(x) =
Good | .3 | $50 | =.3 50
Avg | .5 | $40 | = .540
Weak | .2 | $20 | = .2*20
E(x) = $39
Variance of a Random Variable
- the probability-weighted sum of the squared differences between each possible outcome and the expected value of the random variable
- use financial caluclator
Find the variance of a random variable on the financial calculator
State of Econ. | Prob. | CF. | E(x) =
Good | .3 | $50 | =.3 50
Avg | .5 | $40 | = .540
Weak | .2 | $20 | = .2*20
CF is the X value
Prob is the Y (enter as whole number)
2nd, Data 2nd, clr wrk X01 = $50 Y01 = 30 X02 = $40 Y02 = 50 X03 = $20 Y03 = 20 2nd, Stat 2nd, Set (press until "1 - v") down arrow, need to see N=100 (ie proability = 100) down arrow, X = 39 (ie the expected value of the random variable) down arrow, Sx= 10.49 (sample stdv.) down arrow, Ox= 10.44 (population stdv)
Total Probability rule for expected values =
= E(X) = E(X|S)P(S) + E(X|Sc)P(Sc)
Need to add covariance formulas after doing practice problems
k
Correlation formula =
= row(p) (A,B) = Cov (A,B) / O(A) * O(B)
O is pop stdv
cal O using 2nd data method
Variance of a portfolio formula
= O^2(Rp) = w1^2O1^2 + w2^2O^2 + 2w1w2CovO1*O2
take STDV of answer to get the stdv of the portfolio
Baye’s Formula
=P(E | I)= P(E) * (P(I | E) / P(I))
need to use the total prob rule to solve for P(I)
Combination
- when the order does not matter
nCr = n! / [(n - r)! r!]
Calculator:
n, 2nd, nCr, r
Permutation
- order matters
- nPr = n! / (n-r)!
Calculator
n, 2nd, nPr, r
Labeling
- the number of ways that n items can be labelled with k different labels
n! / [(n1!) * (n2!) * (nk!)
