Probability Flashcards
Random Variable
variable with uncertain outcome.
- Can’t predict what the future outcome is
Ramdom varible example
the exact date of death of a person with insurance
Outcome
Possible value that possible random variable can take.
Outcome example
Can either win or lose at a lottery ticket
Event
SPECIFIED outcome, or set of outcome
Event example
Event A - Return on mutual fund 10 % or less
Event B - if that return is more than 10 %
Tossing a coin 3 times, or 10 % return earned by the portfolio
Mutual exclusivity
I do - to my wife, = I don’t to others.
Only one of them can happen at a time. Can’t have both outcomes at the same time
Exhaustive
cover all possible outcomes
Mutual exclusive example
Coin tossed, it either a head or a tail. Can’t be a head and a tail at the same time.
Exhaustive example
event a - return on a stock is 8 %
event b - return on a stock is < 8 %
event c - return on a stock is > 8%
probability
a number between 0 - 1 that a stated event will happen
Can’t be less than 0 or over a 100 %
Impossible probability - 0
can happen probability - 1
Event impossible probability is ?
if event is impossible probability of 0
example: what is proba that Lebron will mark 100 point in the game - is impossible
Event is certain to happen probaility is ?
if event certain to happen probability of 1
probability example
Portfolio return below 10 %
so a 0.6 probability mean there is a 60% chance that event will happen.
Return on treasury bill - know will get that treasury bill back.
Sum of probability
any set go mutually exclusive and exhaustive events ALWAYS = 1
example - P(a) + P(b) + P(c)= 1
empirical probability (or statistical probability)
based on observations obtained from historical data (probability experience)
empirical probability example
analyzing return data of past 20 years to estimate the return on as tock next year
Subjective probability
based on personal assessment, educated guesses, and estimates
use your opinion to find probability
A priori probability
based on logical analysis and not on personal judgement or observation
A priority probability example
if 2 application for a job, 1 has a 60-70 % chance to get the job, that means the other candidate has a 30-40 % chance to get the job.
Subjective probability example
you think you have an 80% chance of your best friend calling today because her car broke down yesterday and she will need a ride
empirical probability formula
Probability of the event / total probability
total sample of dividend changes = 16,189
frequency of observation that change in dividends is increased = 14,911
frequency of observation that change in dividends is decreased = 1,278
probability that a dividend change is a dividend increase is:
14, 911 / 16,189 = 0.92
ODD FOR event
Based on ratio of the number of ways the vent can occur to the number of ways the event does not occur.
Example, ODD for E = A to b, means: for A occurrences of E, we expect B cases of non-occurence.
ODD example
12 marbles in a bag, 6 green, 4 yellow, 2 bleu.
What is the probability of getting a yellow -> 4/12 = 0.33
What is the ODD for a yellow -> 12 marbles in the bag - 4 yellow = left 8 in the bag.
SO, ODD for yellow is 4/8 = chance or 1 in 2.
ODD AGAINST event
The ratio of the number of ways the event cannot occur to the number of ways the event can occur.
example, OFF Against E = A to B, means:
probability of E = B/(A+B)
Example, Suppose ODD for E = 1 to 7.
Total cases = 1+ 7 = 8
because A= 1, B=7
out of 8 cases, there is 1 case of occurrence and 7 cases of non-occurence.
probability E = 1 / (1+7) = 1/8= = 0.1250
Unconditional probability
other name: Marginal probability
Probability that an event occurs without taking into account other events that happen before.
Stand-alone events
conditional probability
Probability of an event occurring, given that another event has already happened.
P(A|B) -> probability of A, given B.
Even can be > than, = to, < than the unconditional probability depending on the facts.
Unconditional probability example
my basement will flood? the unconditional probability -> probability close to 0, but when rain increase in outcome.
Conditional probability example
What is probability basement will flood, if a hurricane will pass in my area.
Now have hurricane, and flood in basement, those two are linked.
the connection between rain fall and flooding.
Unconditional probability formula - example
The probability that the stock earns a return above the risk free rate (event A)
Probability A = sum of the probabilities of stock returns above the risk free rate / sum of the probabilities of ALL possible returns.
Conditional probability forumula - example
the probability that the stock earns a return above the risk free rate (event A), given that the stock earns a positive return (event B)
P(a|B) = Sum of the probabilities of stock return above the risk free rate / sum of the probabilities for all returns > 0%
Multiplication rule for probabilities
determined joint probability of 2 events occurring -
Joint probability
probability of 2 or more events happening together
Joint probability example
the joint probability of A and B denoted as P(AB) read as the probability of A and B is the sum of the probabilities of their common outcomes.
P(AB) = P(AB)
Multiplication rules for probabilities
A bag contains 16 blue balls and 14 yellow balls.
2 balls are drawn from the bag at random, one after the other without replacement.
What will be the joint probability of both balls being bleu
the probability that the 1st ball is bleu is 16/30.
** find the 30 via (16 + 14)
The probability that the 2ns ball is bleu GIVEN the first one is also bleu, is 15/29
** 29 because took out 1 ball from the bag left with 29 balls. and looking for bleu, and probably found a bleu so its 15 now instead of 16.
So, 15/29 * 16/30 = 0.2759
thus, 27.59 %
Additional rule for probabilities - Mutually exclusive
Add the probably
P(A or B) = P(A) +P(B)
Additional rule for probabilities - NON mutually exclusive
P(A or B) = P(A) + P(B) - P(AB)
Example Candidate asked 2 questions. The probab that she gets the 1st question correst is 0.3, and the proba she gets the 2nd question correct is 0.4
given that the proba that she gets both question correct is 0.1,
what is the probability that she gets either: 1st, 2nd, or both question right?
1ST - 0.3 Pa
2ND - 0.4 Pb
BOTH - 0.1 Pab
We want P(A OR B ) =
0.3 + 0.4 - 0.1 = 0.6
Total probability rule
explain unconditional probability of an event, in terms of that event’s conditional probabilities in a series of mutually exclusive, exhaustive scenarios
diffence between scenario (S) and complementary scenario (S^c)
Example scenario and complementary scenario
Calculating P(A|S)
P(A) = 0.55 P(S) = 0.55 P(S^c)= 0.45 P(A|S^c) = 0.40
P(A) = P(A |S ) * P(S) + P(A |S^c) * P(S^c)
So, 0.55 = P(A|S) * 0.55+ 0.40 * 0.45
move things around
result is 67.27 %
Complement rule
For an event or scenario S, the event non-S is called the complement of A and is denoted as S^C. Since either S or not-S must occur.
P(S) + P(S^C) =1
Example: What is the total probability of the stock rise?
no-recession: 0.7 econ state probability
recession: 0.3 econ state probability
stock performance: rise or fall for no-recession and recession
No recession stock probability rise 0.8, fall 0.2
recession stock probability rise 0.3, fall 0.7
- 8 * 0.7 = 56 %
- 3 * 0.3 = 9%
56 + 9 = 65 %
independent events
independent if the occurence of one event does not affect the probability of the other event.
Independent events example
Playing in the drive way vs will the mail be deliver on time -> nothing to do with each other
correlation of 0
dependent events
two events are dependent when the probability of occurence of one event depends on the occurrence of the other
example dependent events
bags containing 8 green balls and 5 blue balls. If you draw a bleu ball without replacing it, the probability of drawing another bleu ball in your second attempt is greatly changed because you drew a bleu ball the first time.
multiplication rule for independent events - example
P(A) x P(B) X P(C) etc…
suppose the unconditional probability that a fund is a loser in either period 1 or 2 is = 0.50
then period 1 = 0.5
period 2 = 0.5
so, calculating the proba that the fund period 2 loser and the fund is a period 1
then 0.5 x 0.5 = 0.25
Conditional expected value part 1
Values refer to the expected value of random variables X given an event scenario S.
denoted E(X|S)
E(X|S) = P(X1|S)X1 + P(X2|S)X2 …
example: evaluate if the company is a BUY - look at their BS, Executive, estimate cash flow, corp gov, debt level… We think Amazon will have all of this, we think this is what amazon stock price will have. Imagine events outside of amazon like gov aug taxes will affect amazon expected value.
Conditional expected value part 2
- Continuous revision of emptied valus is important in the face of changing investment conditions
- expected value of an investment is affected by the actions of competition, gov, and other financial institutions (example amazon)
total probability rule and the unconditional expected value
Total probability rules is very useful when determining the unconditional expected value of an investment.
example: 20% change the gov will impose tariff on imported cars.
A company assembles cars locally expected return of 14% if the tariff is imposed and return of 11 % if the tariff is not imposed.
what is the unconditional expected return?
with tariff imposed ? 14% x 20%
without tariff (not imposed)? 11% X 80%
** the 80% comes from 100-20% = 80%. whatever was left from the chances.
so, 14x20 +11x80 = 0.0280 + 0.0880 = 0.1160 or 11.60%
tree diagram is
a visual representation of all possible future outcomes and the associated probabilities of random variables
- each diagram represents an outcomes
the expected value of a random variable
is the probability weighted average of the possible outcomes of the random variable
** multiply the probably by each possible outcome
Variance of random variable
expected value of squared deviations from its expected value
Variance >=0
when Variance = 0, there is no dispersion or risk -> the outcome is certain and quantity X is not random at all
The higher the variance, the higher the dispersion or risk, all else equal.
***Difference each possible outcome and substrates from its mean, and square that answer, and take the square root of it to compute the standard deviation
Variance of random varibale formula
sigma^2 (X) = E {(X-E(X))^2}
sigma^2 (X) -> variance of random varibale X
Standard deviation is
- the positive squared root of variance.
- easier to interpret than variance because It is in the same units as the random variable
covariance
measure the degree of co-riskiness or movement between two random variable
how two asset move together
Covariance formula
Cov (Ri,Rf) = sum (P(Ri-ERi) * (Rj-ERf))
covariance interpretation
covariance of return is positive >0
move together return, random variance move together
covariance of return is negative < 0
opposite direction movement. (indirect relationship)
Covariance return is 0 zero when the assets are unrelated.
they are not related - no relationship
Covariance matrix
It s square format of presenting covariance
the off diagonal terms represent variance since COV (Rx,Ry) -> table with the 111 and the 0000
Correlation coefficient
is the ratio of the covariance between two random variables and the product of their two standard deviations
standardized
values lie between -1 and 1
Correlation interpretation
Positive correlation when correlation >0
correlation = (+1)
perfect positive linear relationship
– movement together
negative correlation when correlation < 0
correlation = (-1) perfect negative linear relationship
– inverse relationship
zero correlation
no linear relationship
Correlation (RiRj) formula
(Ri,Rj) = covariance (Ri,Rj) / sigma (Ri) * sigma (Rj)
Expected return on a portfolio
is a weighted average of the expected returns ont he component securities
expected return on a portfolio formula
E(R) = w1 * R1 + w2 * R2 …. etc
variance of a portfolio return
is a function of the variance of the component assets as well as the covariance between each of them
sigma ^2 (Rp) = Wa (weight A)^2 * sigma^2 (Ra) + etc …
Bayes formula
method for updating a probability given additional information.
calculation a probability given for a given even that are updated that reflect current information.
