Probability

\text{Probability}=\frac{\text{Sum of Observations}}{\text{Possibility}}=P(A)=\frac{n(A)}{n(S)}

Review of Set Notation

SSample Space
A=\{a_1,a_2,a_3,…\}
P(S)=1
P(A)\geq1
A\subset BA is a subset of B
\oslashNull or empty set (\oslash is a subset of every set)
A\cup BUnion of A and B, A or B or both
A\cap BIntersection of A and B. In both A and B
A\bar or A^cA compliment (not in A)

Mutually Exclusive A\cap B=\oslash

Laws

Distributive Laws

A\cap (B\cup C)=(A\cap B)\cup (A\cap C)

A\cup(B\cap C)=(A\cup B)\cap (A\cup B)

Associative Laws

(A\cup B)\cup C=A\cup (B\cup C)

(A\cap B)\cap C=A\cap (B\cap C)

Communtative Laws

A\cup B=B\cup A

A\cap B=B\cap A

De Morgan’s Laws

\overline{(A\cap B)}=\bar{A}\cup \bar{B}

\overline{(A\cup B)}=\bar{A}\cap \bar{B}

Conditional Probability and the Independence of Events

P(A|B)=\frac{P(A\cap B)}{P(B)} and P(B|A)=\frac{P(A\cap B)}{P(A)}

A and B are INDEPENDENT if any of the following is true:

P(A|B)=P(A)

(B|A)=P(B)

P(A\cap B)=P(A)P(B)

If DEPENDANT we have P(A\cap B)=P(A)P(B|A)

Laws of Probability

Multiplicative Law

P(A\cap B)=P(A)P(B|A)=P(B)P(A|B)

If independent then P(A\cap B)=P(A)P(B)

Additive Law

P(A\cup B)=P(A)+P(B)-P(A\cap B)

If mutually exclusive then P(A\cup B)=P(A)+P(B)

Compliment

P(A)=1-P(A^c)

P(\overline{A\cap B})=1-P(A\cap B)

P(\overline{A\cup B})=1-P(A\cup B)

P(A|B^c)=\frac{P(A\cap B^c}{P(B^c)}

P(A\cap B^c)=P(B^c|A)P(A)

P(B^c|A)=1-p(B|A)

P(B^c|A^c)=\frac{P(B^c\cap A^c)}{P(A^c)}

Counting Sample Points

Multiplication Principle or mn rule: E.g. There are 4 suits and 13 cards per suite in a deck of playing cards. The number of distinct cards are therefore 4*13=52.

Combinations: n objects taken r at a time (the order is irrelevant). The calculation is called “n choose r“.

{n \choose r}=C^n_r=\frac{n!}{r!(n-r)!} E.g. {150 \choose 3}=\frac{150!}{3!(150-3)!}=\frac{(150 \cdot 149 \cdot 148)(147 \cdot 146 \cdot \cdots \cdot 1)}{(3 \cdot 2 \cdot 1)(147 \cdot 146 \cdot \cdots \cdot 1)}=\frac{(150 \cdot 149 \cdot 148)}{(3 \cdot 2 \cdot 1)}

Permutations: An ORDERED arrangement of r distinct objects. The number of ways of ordering n distinct objects r at a time.

P^n_r=n(n-1)(n-2) \ldots \(n-r+1)=\frac{n!}{(n-r)!}

Partitioning: Number of ways to pertition nn distinct objects into k distinct groups (each object only in one group).

N={n \choose n_1n_2 \cdots n_k}=\frac{n!}{n_1!n_2! \cdots n_k!}