Probability Theory

Question 1

What is an event?

Answer 1

is a subset of the sample space, i.e. one or multiple possible out-
comes. An event consisting of a on or more element in Ω

Question 1

What is part of the Probability Space (Ω, F, P ), and what is the definition of each mathematical concepts?

Answer 1

• Sample space Ω, that is the set of all possible elementary outcomes of an experiment.
• The Sigma field (σ-field) F over Ω, is the set of all possible events that we consider in an experiment, the set of all possible combinations [or,and,not] of elementary outcomes as well as ∅ (empty event) and Ω (full set of events) wich means that the event has no/any outcome.
• Probability measure P that assigns a probability to each event

Question 2

What are the 3 set operations that can be used on events?

Answer 2

• A ∩ B, intersection of A
  and B, ( if A ∩ B =0 A and B are mutually exclusive, or disjoint)
  -A ∩ B, which is A, or B, or both, and is called the union of A and
  B
• Ac is the event of ‘not A’, and is called the complement of A.

Question 3

What are the properties of the sigma-field F?

Answer 3

1. Ω ∈ F (‘The experiment has an outcome’ is an event)
2. If A is an event, then ‘not A’ is also an event
3. If A1 and A2 are events, then ‘A1 or A2’ is also an event.

Question 3

What are the properties of the probability measure P ?

Answer 3

1.P (∅) = 0: the probability of the empty event is 0.
P (Ω) = 1: the probability of the sample space is 1.

2. for all events A, P (A) ≥ 0
3. for disjoint events A and B, P (A ∪ B) = P (A) + P (B). Probability is
   additive for mutually exclusive events.
4. A ∪ Ac = Ω and A ∩ Ac = ∅, we have p(Ac) = 1 − P (A) (or
   the probability that an event does not happen is 100% minus the probability
   that it happens)

Question 4

What is the definition of conditional probability P (A|B) ?

Answer 4

The probability that A occurs given that B has occurred

Question 5

What is the formula of the conditional probability?

Answer 5

P (A|B) = P (A ∩ B)/P (B)

if P (B) > 0
undefined if P (B) = 0

The sample space is reduced to B

Question 6

Explain the Theorem of total probabilities

Answer 6

If a sample space Ω is partitioned in events A that are mutually exclusive (Ai ∩ Aj = ∅) and collectively exhaustive (their sum adds up to 1n
U from i=1to n of Ai = Ω), then given an event B, P(B)= tot the sum of each Ai intersection B event that can be rewritten using the conditional formula as P(B|Ai)P(A)

Question 7

When are two event A and B mutually exclusive or disjointed?

Answer 7

When A ∩ B=0 or B ∩ A =0

Question 8

When are two events A and B independent? How does the conditional probability of the event A and B change?

Answer 8

P (A ∩ B) = P (A) P (B)

if A and B are independent events then it follows that P (A|B) =P (A).

(Indeed, knowing that B has happened does not change the probability for A)

INDEPENDENCE IS A STRONG ASSUMPTION, BE CAREFUL MAKING IT!

Question 9

What is a uniform probability measure?

Answer 9

Calculating a probability with the assumption that each outcome is equally likely.

In this case for an event A:
P(A) = |A|/|Ω| = # outcomes in A/total # of outcomes

Question 10

Explain Bayer’s Law (write the formula for the conditional probability P(A|B) and P(B|A) and th formula in the case it is used to test a specific case Ai of a partition Ai

Answer 10

P (A|B) = P (B|A)P (A)/P (B)

and

P (Ai|B) = P (B|Ai)P (Ai)/
P (B|A1)P (A1) + . . . + P (B|An)P (An)

Where the denominator comes from the multiplicative rule

P(A∩B) = P(B|A)P(B)