C01 FUNDAMENTALS OF PROBABILITY

BASIC TERMS  #

• Probability—Chances of happening or non happening of any event 
• A random variable is an uncertain quantity/number. 
• An outcome is an observed value of a random variable. 
• An event is a single outcome or a set of outcomes.

CONDITIONAL PROBABILITY  #

Unconditional probability P(A):- Also known as marginal probability is the one which dose not depend on any event (Eg. as stated in our lecture:- You buying Dish TV or Videocon D2H. 

Conditional Probability P(A|B) :-  Occurrence of one event affects probability of other (eg. You buying Dish TV conditional upon Buying Samsung TV), Denoted as P(A|B) = Probability of A (Buying Dish TV) Given (|) B (Buying Samsung). 

Joint Probability (AB) :-  (Multiplication Rule) Probability that both event will occur, P(AB)= P(A|B) X P(B). 

INDEPENDENT AND MUTUALLY EXCLUSIVE EVENTS  #

Independent event:- occurrence of one event has no influence on other event: events are independent only if p(a|b) = p(a) or p(b | a) = p(b). 

If these conditions are not satisfied then conditions are said to be dependent events 

Mutually exclusive events:-  happening of one event eliminate the possibility of happening of other event then both events are said to be mutually exclusive events. Hence joint probability will be zero between two  

P(ab) = 0. 

Remember the difference between two types of events :- you are likely to see question based on these two concepts in story form 

BAYES’ RULE  #

Allows us to use information of outcome of one event to improve our estimates of the unconditional probability  of another event.  

P(A|B) = P(B | A)XP(A) / P(B) 

P(A) and P(B) are total probabilities 

PDF AND CDF  #

Probability function :- p(x) is probability that random X will take value of x P(X=x) = P(x) 

Two key properties of probability function  

0 ≤ p(x) ≤1 & total of p(x) = 1 

Continuous random variable :-  No of possible outcomes infinite. 

PDF probability density function= denoted as f(x) – this is derivative of CDF (cumulative distribution function) denoted as F(x)( capital F here) 

Note- This concept is directly linked to chapter no 3 of quants Distribution hence we will cover all the aspects in that chapter. 

VARIOUS CALCULATIONS IN PROBABILITY  #

Probability that at least one of two events will occur:-  Addition rule of probability:-  In this case we are looking for probability that at least A or B  or both A and B will occur. 

A. Events are not mutually exclusive event:-  

     P (A or B) = P(A) + P(B) – P(AB)  

Due to double counting of P(AB), we are subtracting once so double counting problem can be eliminated. 

So if we want to calculate probability that either A or B will happen but not both then we have to reduce complete portion of joint probability. Hence formula becomes  

P( only A or B but not both) = P(A) + P(B) – 2 X P(AB) 

= P(A or B) – P(AB) 

B. Mutually exclusive events:-   When events are mutually exclusive then there is no chance of common area or say joint probability between both  which result in P(AB) =  0.  Hence in that case

     P (A or B) = P(A) + P(B)  

Calculating Joint Probability of any number of events :- 

On Role of two dice, “Joint probability” of getting two 4’s = 1/6 X 1/6 = 1/36 

As events are independent we can apply simple multiplication rule 

Calculation of Joint probability in case of independent events :-  

Probability of A and B = P (A) X P (B) 

Probability of A or B =  P(A) + P( B)  

School level math:-  

P(A and B) Denoted as P(A n B) 

P(A or B) Denoted as P(A U B)