2. Basic Probability Flashcards
What is a Measure?
Definition 12. Given a sample space (or universal set) S, a (finite) measure is a set function that assigns a non-negative real number m(A) to every set A ⊆ S, which satisfies the following condition: for any two disjoint subsets A and B of S (ie A ∩ B = ∅),
m(A ∪ B) = m(A) + m(B).
Properties of a measure. (M1 - M6)
(M1) If B ⊆ A, then m(A \ B) = m(A) − m(B).
(M2) If B ⊆ A, then m(B) ≤ m(A).
(M3) m(∅) = 0
(M4) m(A ∪ B) = m(A) + m(B) − m(A ∩ B).
(M5) For any constant c > 0, the function g(A) = c×m(A) is also a measure.
(M6) If m and n are both measures, the function h(A) = m(A) + n(A) is also
a measure.
What is a Partition on a set?
A partition of a set S is a collection of sets E = {E1, E2, . . . , En} such that
1. Ei ∩ Ej = ∅, for all 1 ≤ i, j ≤ n with i 6= j;
2. E1 ∪ E2 ∪ . . . ∪ En = S.
We say that {E1, E2, . . . , En} are mutually exclusive and exhaustive.
Describe a Measure on a Partition of a set.
The measure of a set is the same as the sum of the measures of each subset in the sets partition.
Define Probability as a Measure
Suppose S is a sample space and P a measure on S. We say that P is a probability (or probability measure) if P(S) = 1. Hence probability is defined to be a special case of measure: a measure that takes the value 1 on a sample space S.
What is a Permutation? (nPr formula)
A permutation is the number of ways of choosing r elements out of n, where the order matters.
nPr = n! / (n-r)!
What is a Combination? (nCr formula)
A permutation is the number of ways of choosing r elements out of n, where the order doesn’t matter.
nCr = n! / r!(n-r)!
Define Fair Odds
Informally, we define fair odds to mean odds that you think favour neither the bookmaker or the gambler. If you think the odds favour the gambler, you’d expect the bookmaker to make a loss. If you think the odds favour the bookmaker, you’d expect the bookmaker to make a profit.
What are the odds on and odds against an event?
For an event E, the odds on the event E are P(E) / 1 − P(E), and the odds against the event E are 1 − P(E) / P(E).
Formula for conditional probability P(E|F)
P(E|F) = P(E and F) / P(F)
Independent Events
Two events E and F are said to be independent if
P(E ∩ F) = P(E)P(F), or if P(E|F)=P(E).
The law of total probability
Suppose we have a partition E =
{E1, . . . , En} of a sample space S. Then for any event F,
P(F) = (sum from 1 to n) P(F ∩ Ei), or, equivalently, P(F) = (sum from 1 to n) P(F|Ei)P(Ei).
Bayes Theorem
Suppose we have a partition of E = {E1, . . . , En}
of a sample space S. Then for any event F,
P(Ei|F) = P(Ei)P(F|Ei) / P(F).
This leads to,
P(E|F) = P(F|E)P(E) / (P(F|E)P(E) + P(F|E¯)P(E¯)).
What is the power set of A?
The set of all subsets of A.
