# Basic Probability and Set theory

Basic Set theory:

• A is a proper subset of B <=> $A \subset B$ and $A \neq B$
• A = B <=> $A \subset B$ and $B \subset A$
• A and B are said to be disjoint or mutually exclusive iff $A \cap B = \phi$

De Morgans Law:

• $(A \cup B )^c = A^c \cap B^c$
• $(A \cap B )^c = A^c \cup B^c$
• $(A^c \cup B^c ) = (A \cap B)^c$
• $A = A \cap B^c$
• $B = B \cap A^c$

Basic Probability:

• $(C_{r}^{n})= \frac{n!}{r!(n-r)!}$
• $(P_{r}^{n} ) = (P_{r-1}^{n-1}) + (P_{r}^{n-1})$
• $(x + y)^{n} = \sum_{k=0}^{n}x^{k}y^{n-k}$
• No. of non negative integer solutions to $x_1 + x_2 + x_3 + …. x_n = r$ : $(_{r}^{n+r-1} )$. [ # of ways to distribute r identical objects into n distinct boxes = # of ways to select r objects w/ repetition from n different types of objects]

Axioms of Probability:

A set function that associates a real value P(A) with the event A is called the probability of A, if the following axioms are satisfied:

• $0 \leq P(A) \leq 1$
• $P(S) = 1$
• $P(\bigcup_{i=1}^{\infty }A_{i}) = \sum_{i=1}^{\infty}P(A_{i}) where A_{i} \cap A_{j} = \phi$

Unions and Intersections

• $P(A \cup B) = P(A) + P(B) – P(A \cap B)$
• $P(A \cup B \cup C) = [ P(A) + P(B) + P(C)] \ – [P(A \cap B) + P(B \cap C) + P(C \cap A)] \ + [P(A \cap B \cap C) ]$
• $P(A \cup B \cup C \cup D) = [ P(A) + P(B)+ P(C) + P(D)] \ – [P(A \cap B) + P(B \cap C) + P(C \cap A) + P(A \cap D)]+ P(B \cap D) + P(C \cap D) + [[P(A \cap B \cap C) + P(A \cap B \cap D) + P(B \cap C \cap D) + P(A \cap C \cap D)]\ – [P(A \cap B \cap C \cap D)$ [ i.e for 4 events, $(_{1}^{4})$ summants – one each simple event, $(_{2}^{4})$ summants – pair of events, $(_{3}^{4})$ summants – Three events, $(_{4}^{4})$ summants – 4 events. Alternate signs]

Boole’s Inequality:

If $A_1, A_2, A_3$….are a sequence of events then $P(\bigcup_{i=1}^{\infty }A_i) \leq \sum_{i=1}^{\infty}P(A_i)$
For finite unions, $P(\bigcup_{i=1}^{k }A_i) \leq \sum_{i=1}^{k}P(A_i)$
The RHS provides an upper bound when $\sum_{i=1}^{k}P(A_i) < 1$

Bon Ferroni’s Inequality:
If $A_1, A_2, A_3 ….A_k$are a sequence of events then $P(\bigcap_{i=1}^{k }A_i) \geq 1 – \sum_{i=1}^{k}P((A_i)^c)$
For finite unions, $P(\bigcup_{i=1}^{k }A_i) \leq \sum_{i=1}^{k}P(A_i)$
The RHS provides an upper bound when $\sum_{i=1}^{k}P(A_i) < 1$

Conditional Probability:

The conditional Probability of an event A given that the event B has occurred is denoted by $P(A/B) = \frac{P(A \cap B)}{P(B)}$
Since $1 / P(B)$ is > 1, Conditional probability would satisfy the 3 axioms of the reduced sample space.
$P(A \cap B) = P(A).(B/A)$
$P(A \cap B \cap C) = P(A).(B/A).P(C/A \cap B)$
$P(A_1 \cap A_2 \cap A_3 ….. \cap A_n) = \ P(A_1).P(A_2/(A_1 \cap A_2)) …..P(A_n/(A_1 \cap A_2 \cap A_3 …. \cap A_n))$

Independent Events:

$P(A \cap B ) = P(A). P(B)$
$P(A /B ) = P(A)$
$P(B /A) = P(B )$
If two events A and B are independent then the following pairs are also independent (i) $A^c and B^c$ (ii) $A^c and B$ (iii) $A and B^c$

Baye’s theorem:

Let $A_1, A_2, A_3, ….A_k$ be k mutually exhaustive and exclusive events. For any normal event B   $P(B) = P(A_1).P(B/A_1) + P(A_2).P(B/A_2) + …. + P(A_k).P(B/A_k)$