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Basic Probability and Set theory

  • SSP 

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   $P(B) = P(A_1).P(B/A_1) + P(A_2).P(B/A_2) + …. + P(A_k).P(B/A_k)$

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