QM PREREQ3 – Probability Concepts

Question 1

How are probabilities estimated?

Answer 1

Empirical probabilities are based on historical observation. The past is assumed to be representative of the future (not necessarily true). The historical period must include occurrences of the event

Subjective probabilities involve adjusting an empirical probability based on an intuition or experience.
We may do this when there is a lack of empirical observations, or to make a personal assessment

A priori probabilities involve arriving at a conclusion based on deductive reasoning. I.e., if a die has 6 sides the probability of rolling 6 is 1/6. This is perhaps the most objective method of estimation

Question 2

How are probabilities stated?

Answer 2

2 ways:
if there is a 10% chance:

odds:
1 to 9 chance (for every 1 occurence we expect 9 non-occurence)
9 to 1 chance of non occurence.
probability:
1 in 10 chance (out of 10 instances we expect 1 occurence)

Question 3

What are conditional and unconditional probability?

Answer 3

Unconditional probabiltiy is P(A)
Conditional probability is P(A │ B)

Probability of A occuring given B
We could also illustrate this as a venn diagram. Conditional probability is the intersection between the A circle and B circle.
Unconditional probability would just be the value of the whole A circle

The mhltiplication rule means that P(AB) = P(A│B) x P(B)

Also therefore P(A│B) = P(AB) / P(B)

Question 4

What is complement?

Answer 4

A and A complement is the probability of A + the probability of not A
Therefore A + A complement = 1

Question 5

What is the addition rule?

Answer 5

P (A or B) = P (A v B) = P(A) + P(B) - P(AB)
Since we are double counting otherwise

Question 6

What are dependent and independent events?

Answer 6

2 events are independent iff:
P (A│B) = P(A)
or P(B│A) = P(B):
Knowing B tells us nothing about A

A dependent event is where P(A) is related to P(B)
e.g., A = stock Q rises
B = SP500 rises
A is most likely dependent on B

Question 7

What is the difference between combination and permutation?

Answer 7

A combination is the number of ways of selecting r objects from n where order does not matter
nCr = (^n r) = n! / ((n - r)! x r!)

A permutation is the number of ways of selecting r objects from n where order does matter
nPr = n! / (n - r)!
So we don’t divide by r! for permutations since there are more possibilites

A recombining lattice is like a probability tree that joins up
This is used for permutations
And in FA in asset price moves