Probability (Part-2)

Binomial Distribution : 

                                   The imposter distribution and two binomial distributions ...
As the name suggests , in this distribution only two outcomes are possible like 0 or 1 , Success or Failure , Yes or No etc. These type of distributions are known as binomial distribution. It is applicable when an event is occurring multiple times for example we toss coin 20 times, here the possibility of outcomes of coin is either head or tail and it occurs 20 times so here, we will use binomial distribution. Bernoulli Distribution is also same as that of binomial distribution, it also has two outcomes possible.

Multinomial Distribution : 

                                     Multinomial probability density function - MATLAB mnpdf

In this probability distribution, we get the outcomes from the multiple experiments for the occurrence of the particular event. As the name suggests, in this this type of probability distribution, the experiment is consists n trials and we get the probability trials.
Suppose a multinomial experiment consists of n trials, and each trial can result in any of k possible outcomes: E1, E2, . . . , Ek. Suppose, further, that each possible outcome can occur with probabilities p1, p2, . . . , pk. Then, the probability (P) that E1 occurs n1 times, E2 occurs n2 times, . . . , and Ek occurs nk times is:

P = [ n! / ( n1! * n2! * ... nk! ) ] * ( p1n1 * p2n2 * . . . * pknk )

where n = n1 + n2 + . . . + nk. 

Conditional Probability : 
 
                   Conditional Probability - A Plus Topper
In this, we solve the probability of an element with respect to the given condition or the evidence provided. If A,B C S and  P(B) > 0 then the probability of A given B is given by 
                                   P(A|B) = P(B|A)/P(B)

Example : You roll a dice.What is the P(A|B) if A = {6 appears} and B = {the outcome is even}.

P(A|B) /P(B) =(1/6)/(1/2) = 1/3

Law of Total Probability : 

          
                 Conditional Probability - Law of Total Probability — The Tech Pro

 if  is a finite  of a Sample space (in other words, a set of pairwise disjoint events whose union is the entire sample space) and each event  is measurable, then for any event  of the same probability space:


Bayes Rule : Under same condition as law of total probability 
                      P(Bj|A) = P(A|Bj)*P(Bj)/ sum of P(A|Bj)*P(Bj)


Independence : A and B are called independent if P(A ∩ B) = P(A)P(B). If P(B)>0, this is equivalent to P(A|B) = P(A)

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