chapter 2

Permutation

An ordered subset is called a permutation. The number of permutations of size k that can be formed from the n individuals or objects in a group will be denoted by Pk,n

Combination

An unordered subset is called a combination. One way to denote the number of combinations is Ck,n, but we shall instead use notation that is quite common in probability books: (n k), read "n choose k."

Proposition 2

For any event A, P(A) + P(A') = 1, from which P(A) = 1 - P(A')

Proposition 3

For any event A, P(A) ≤ 1

Axiom 1

For any event A, P(A) ≥ 0 the chance of A occurring should be non negative

For any two events A and B with P(B) > 0, the conditional probability of A given that B has occurred is defined by

P(A I B) = P(A ∩ B)/P(B) B is the "conditioning event."

The Multiplication rule

P(A ∩ B) = P(A I B)P(B)

The multiplication rule for P(A ∩ B)

P(A ∩ B) = P(A|B) P(B) = P(A)P(B) A and B are independent if and only if (iff) P(A ∩ B) = P(A)P(B)

Two events A and B are independent if

P(A|B) = P(A) and are dependent otherwise. It is then natural to regard A and B as independent events, meaning that the occurrence or nonoccurrence of one event has no bearing on the chance that the other will occur. if two events are mutually exclusive, they cannot be independent.

Axiom 2

P(S) = 1 the maximum possible probability of 1 is assigned to S

Proposition 1

P(∅) = 0 where ∅ is the null event (the event containing no outcomes whatsoever). This in turn implies that the property contained in Axiom 3 is valid for a finite collection of disjoint events.

Proposition 6

Pk,n = n! /(n - k)!

Bayes' Theorem

The probability of an event occurring based upon other event probabilities.

Intersection

of two events A and B, denoted by A ∩ B and read "A and B," is the event consisting of all outcomes that are in both A and B

Mutually exclusive

two or more events that cannot happen simultaneously, also known as disjoint events

Complement

of an event A, denoted by A', is the set of all outcomes in S that are not contained in A. (not A)

Union

of two events A and B, denoted by A U B and read "A or B," is the event consisting of all outcomes that are either in A or in B or in both events (so that the union includes outcomes for which both A and B occur as well as outcomes for which exactly one occurs)—that is, all outcomes in at least one of the events

Null event

the event consisting of no outcomes whatsoever, denoted by ∅. When A ∩ B = ∅, A and B are said to be mutually exclusive or disjoint events

Sample space

the set of all possible outcomes of that experiment, denoted by S

Probability

the study of randomness and uncertainty

Mutually independent

The events are mutually independent if the probability of the intersection of any subset of the n events is equal to the product of the individual probabilities.

Law of Total Probability

The probability of an event is the sum of its probability across every possible condition

Sum of a geometric series

a + ar +ar^2 +ar^3 + ... = a/(1-r)

Proposition 7

(n k) = Pk,n / k! = n! /k!(n - k)!

Proposition 4

For any two events A and B, P(A U B) = P(A) + P(B) - P(A ∩ B) When events A and B are mutually exclusive, then P(A U B) = P(A) + P(B) For any three events A, B, and C, P(A U B U C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) -P(B ∩ C) + P(A ∩ B ∩ C) In general, the probability of a union of k events is obtained by summing individual event probabilities, subtracting double intersection probabilities, adding triple intersection probabilities, subtracting quadruple intersection probabilities, and so on

Axiom 3

If A1, A2, A3,... is an infinite collection of disjoint events, then P(A1 U A2 U A3 U ...) = Σ P(Ai) if we wish the probability that at least one of a number of events will occur and no two of the events can occur simultaneously, then the chance of at least one occurring is the sum of the chances of the individual events

Proposition 5 (product rule)

If the first element or object of an ordered pair can be selected in n1 ways, and for each of these n1 ways the second element of the pair can be selected in n2 ways, then the number of pairs is n1n2

Product rule for k-tuples

Suppose a set consists of ordered collections of k elements (k-tuples) and that there are n1 possible choices for the first element; for each choice of the first element, there are n2 possible choices of the second element;...; for each possible choice of the first k -- 1 elements, there are nk choices of the kth element. Then there are n1n2 ... nk possible k-tuples

Experiment

any activity or process whose outcome is subject to uncertainty

Event

any collection (subset) of outcomes contained in the sample space S. An event is simple if it consists of exactly one outcome and compound if it consists of more than one outcome