# Definitions

- Set
- Collection of related objects, such that it's possible to determine for any given object whether or not it belongs in the set.
- Subset
- If every element of the set A is in the set B, then A is a subset of B.
- Universal Set
- The set of all elements (in any given context).
- Empty Set
- The set containing nothing. (Subset of any set).
- Complement
- The set containing everything not in the set A.
- Union
- The set containing everything in A, and everything in B.
- Intersection
- The set containing only what is in A
**and**B. - Difference
- The set of A - B is defined as the
*intersection*of A and B^{c} - Mutually Exclusive / Disjoint
- Two sets are mutually exclusive if they contain no elements. I.e. if the union of A and B is the Empty Set.
- Sample Space
- The set of all possible outcomes of a given experiment.
- Sample Point / Outcome
- Any one possible outcome in a sample space.
- Event
- An event E is a subset of a sample space.
- Statistically Independent
- If P(A Intersection B) = P(A)P(B).
- Random Variable
- A real valued function defined on a sample space.
- Probability Distribution (of a random variable X)
- A description of all the probabilities of all events associated with X.
- Binomial Distribution

# Probability

- Probability
- A function performed on a sample space such that it fits four conditions:

- 0 <= P(A) <= 1
- P(∅) = 0
- P(S) = 1
- If A and B are mutually exclusive events then P(A Union B) = P(A) + P(B).

## Addition Rule

Three rules that apply to all events:

- P(A Union B) = P(A) + P(B) - P(A Intersection B)
- P(A
^{c}) = 1 - P(A) - If A is a subset of B then P(A) <= P(B)

## The Three Counting Rules

- If there are
*k*experiments, each with*n_i*possible outcomes then the total number of possible outcomes for the*k*experiments becomes the product of each*n_i*multiplied together. - The number of possible permutations of
*r*objects selected from*n*distinct objects is $^nP_r = \frac{n!}{(n-r)!}$ - The number of ways of choosing r objects from n distinct objects is $^nC_r = \frac{n!}{r!(n-r)!}$

## Conditional Probability

The probability of A given B is the probability of (A *and* B) over probability of B.

## Total Probability Rule

If B is an event in S, and A_i is a partition of S then:

(2)## Bayes' Rule

If B is an event in S, and A_i is a partition of S then:

(3)## Probability Distribution

The probability distribution for a discrete random variable X where ${X_k : k \in \mathbb{Z}}$ is a set ${p_k : k \in \mathbb{Z}}$ such that:

(4)I.e. the probability of any possible result of the function is included in the p set, at the corresponding location. And, of course, the sum of all the probabilities is 1 (by definition).

### Expected Value

The **expected value** for X is shown as:

### Variance

The **variance** of X is shown as:

### Standard Deviation

The **standard deviation** of X is shown as:

### Binomial Distibution

If we have n trials and the probability of success for any trial is *p* and hence the probability of failure, *q* is 1 - p, then:

If X is a random variable with Binomial distribution B(n,p) then X:

a) has a mean of *np*

b) has variance of *npq*