Study Session 2 - Reading 8 Probability Concepts

LOS 8a - Define a random variable, an outcome, an event, mutually exclusive events, and exhaustive events
These are fairly intuitive, so I'll just deal with the last two:

• mutually exclusive = cannot both happen at the same time
• exhaustive = includes all possible outcomes (probabilities will sum to 1)

LOS 8b - Explain the two defining properties of probability and distinguish among empirical, subjective and a priori probabilities

• Probability is always between 0 and 1
• Sum of probabilities of mutually exhaustive and mutually exclusive events will be 1.
  
• empirical probability is established through analysis of past data
• a priori probability is deduced or uses logic e.g. 5 out of 10 stocks yesterday were up therefore a random stock from this ten had a 50% probability of going up
• subjective = educated guess

LOS 8c - State the probability of an event in terms of odds for or against the event

• Here I rely on how i hear this in regular speech. e.g. if the odds are 9 to 1 against an event, I know there is a good chance (90%) that this will not occur.
• Only tricky part is remember that the above would not be 10 to 1. e.g. if probability of an event happening is 20% then the odds in favour are 1 to 4 and the odds against are 4 to 1.

LOS 8d- Distinguish between unconditional and conditional probabilities

• Unconditional = prob of an event regardless of past or future occurrence of other events
• Conditional = occurrence of one event affects the probability of the occurrence of another event

LOS 8d - Define and Explain the multiplication, addition, and total probability rules

• mult. rule = P(AB) = P(AB) * P(B) i.e. prob of A and B taking place where A is dependent on B
• addition rule = P(A or B) = P(A) + P(B) - P(AB) i.e. prob of A or B taking place is the sum of their probabilities minus the prob of both taking place (to avoid double counting). NB don't forget to subtract the overlap
• total prob = P(AB₁)P(B₁) + P(AB₂)P(B₂) +...+ P(AB_n)P(B_n)

LOS 8f: Calculate and interpret (1) the joint probability of two events, (2) the probability that at least one of two events will occur, given a probability of each and the joint probability of two events, and (3) a joint probability of any number of independent events

1. joint probability of two events A,B is calc. using mult. rule P(AB) = P(AB)*P(B)
2. if the events are independent, then P(AB) = P(A) so joint prob is P(AB) = P(A)*P(B)
3. prob. of at least one of two events occuring is addition: P(A or B) = P(A)+P(B)-P(AB)

LOS 8g: Distinguish between dependent and independent events

This is fairly intuitive. Events A and B are independent IFF:
P(AB) = P(A) or vice versa, P(BA) = P(B)
otherwise they are dependent

LOS 8h: Calculate and interpret, using the total probability rule, an unconditional probability

The total probability (unconditional probability) of an event R is calc.:

P(R) = P(RS₁) * P(S₁) + P(RS₂) * P(S₂) +...+ P(RS_n) * P(S_n) where the set of events {S₁), S₂),...S_n)} is mutually exclusive and exhaustive

This is also fairly intuitive in a real world setting. Say you are trying to find the total probability of a rise in interest rates given the state of the economy:

P(Poor Economy) = 0.30
P(Interest Rates Rising Poor Economy) = 0.10
P(Normal Economy) = 0.50
P(Interest Rates Rising Normal Economy) = 0.40
P(Good Economy) = 0.20
P(Interest Rates Rising Good Economy) = 0.70

The total probability is the sum of the joint probability for each event i.e.
(0.30)(0.10)+(0.50)(0.40)+(0.20)(0.70) = 0.03+0.20+0.14 = 0.51 or 51% probability

Note: This will often be structured as a tree diagram which is a good way to visualise it if they do not present it this way and helps visual people like me "get it"

Expected Value

The degree of dispersion of outcomes around an expected value of a random variable is measured using the variance and standard deviation

When pairs of random variables are being observed, the covariance and correlation are used to measure the extent of the relationship between the observed values for the two variables from one obs to the next

The Expected Value is the weighted average of all the possible outcomes of a random variable (e.g. interest rate rise) where the weights are the probabilities that the outcome will occur. Again this is fairly intuitive: it is the sum of the final node results in the tree diagram:

E(X) = P(x1)x1 + P(x2)x2 + ... + P(xn)xn

NB when all probabilities are equally likely, the E(X) is simply the arithmetic mean

LOS 8i: Explain the use of conditional expectation in investment applications

Just like the interest rates example above, conditional expected values are contingent upon the outcome of some other event (e.g. state of the economy). Conditional expected value would be revised using Bayes' formula when new information arrives.

LOS 8j: Diagram an investment problem using a tree diagram

This is a visual representation of the kind of problem shown above. Say a stock price moves with the state of the economy (up or down). We would have four cases:

1. good economy, stock up
2. good economy, stock down
3. bad economy, stock up
4. bad economy, stock down

As in the example with interest rates, each comes with its own probability (e.g. prob. of a good economy and then corresponding prob. of stock going up and prob. of stock going down)

**********Must update this to include tree diagram***********

LOS 8k: Calculate and interpret covariance and correlation

Cov(Ra,Rb) = Sum of Prob(x) * (dev from mean for A for scenario x)(dev from mean for B for scenario x)

So this is calculated in four steps:

1. calculate expected return for A and B
2. calculate deviations from expected return for A and B for each scenario
3. multiply the deviations for A and B for each scenario and then multiply the product by the probability for that scenario
4. sum the resulting products

Correlation
Correlation is an easier to interpret measure of the same relationship between A and B and is found by:

Correlation(A,B) = Cov(A,B)/σA * σB

Correlation properties:
ranges from -1 to +1 (perfect negative correlation to perfect positive correlation with zero being no correlation)

LOS 8l: Calculate and interpret the expected value, variance and standard deviation of a random variable and of returns on a portfolio

Portfolio expected value = weighted average of the assets (or returns)

Portfolio variance took me a while to process because it looks scarier than it actually is. It is simply a way of taking the sum of the weighted variances of each asset's return and the weighted covariance of the assets' returns. So for assets A and B the formula is:

Var(Rp) = w_A²σ_A² + w_B²σ_B² + 2w_Aw_Bσ_Aσ_BCorr(A,B)

• Note: σ_Aσ_BCorr(A,B) is the covariance(A,B) so if you are not given the Corr, you can find it for the first half of the formula and then substitute the Cov(A,B) after the weights in the second half. Also, this is for a portfolio of only two stocks.
• For more stocks, the number of the weighted variances increases by the number of stocks and the number of Cov increases such that all possible pairings of stocks are considered.

LOS 8n: Calculate and interpret an updated probability using Bayes' formula

Well, this one really messed me up. As usual, it was simpler than I assumed by looking at the formula. The definition for the formula is:

updated probability = (prob. of new info for given event/uncond. prob. of new info)*prior prob. of event

This didn't make sense to me because i couldn't tell what the new info was and what the unconditional info was etc.

The way I do it is by thinking of the tree diagram. If we diagram out the possibilities for a stock rising or falling given different probabilities for different states of the economy and then we are told that the stock did rise and we are asked for the probability that the economy was good as a result then my formula would be:

updated probability = probability that stock rose and econ was good/total prob. of economy being good

That is the resulting prob at the end of the node where the stock rose and the econ was good divided by the sum of the resulting probabilities at the end of all the nodes where the econ was good.

Bayes tells us the updated probability now that we know the "answer" to the original question i.e. that the stock rose

LOS 8o: Identify the most appropriate method to solve a particular counting problem and solve counting problems using the factorial, combination and permutation notations

The way this was described was confusing but the bottom line is that you are trying to figure out how many options there are to assigning a set of data to different groups such as if you were trying to assign employees to different development teams.

• If the number of groups is the same as the number of people then use n! (n[2nd][x!])
• If the number of groups is not the same and order is not important then n [2nd][nCr] r where n is the total number and r is the number of groups n!/(n-r)!r!
• If the number of groups is not the same and order is important then n [2nd][nPr]r where n is the total number and r is the number of groups n!/(n-r)!