1.1.2 Probability Theory

Question 1

What is probability?

Answer 1

A measure of how likely an event is to occur, represented by a number between 0 and 1.

Question 2

How do we denote the probability of an event A?

Answer 2

P(A), where A is the event.

Question 3

What are the three basic properties of probability?

Answer 3

1. 0 ≤ P(A) ≤ 1 for any event A.
2. P(∅) = 0 (the probability of an impossible event is 0).
3. P(S) = 1 (the probability of the entire sample space is 1).
4. The union of all events is equal to their sum

Question 4

What is the Additive Law of Probability?

Answer 4

For any two events A and B:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
If A and B are mutually exclusive, then:
P(A ∪ B) = P(A) + P(B)

Question 5

What is the formula for the union of three events?

Answer 5

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(A ∩ C) + P(A ∩ B ∩ C)
This corrects for overcounting intersections.

Think venn diagrams

Question 6

What is the Law of Total Probability?

Answer 6

If B1, B2, …, Bn form a partition of the sample space, then for any event A:
P(A) = P(A ∩ B1) + P(A ∩ B2) + … + P(A ∩ Bn)
or rewritten using conditional probability:
P(A) = P(A | B1) P(B1) + P(A | B2) P(B2) + … + P(A | Bn) P(Bn)

Question 7

What are De Morgan’s Laws?

Answer 7

1. (A ∪ B)c = Ac ∩ Bc (the complement of a union is the intersection of complements).
2. (A ∩ B)c = Ac ∪ Bc (the complement of an intersection is the union of complements).

Question 8

true or false? if A is a subset of B then P(A) is equal to or less than P(B).

Answer 8

true

Question 9

what is the special case of law of total prob/total prob thrm where there are only two partitions?

Answer 9

P(B) = P(B intersect A) + P(B intersect A^c)