Binomial Probability- Suppose that a trial or an experiment whose outcome can be classified as either success or failure is performed.
Suppose now that n independent rials , each of which results in a success with probability p and in a failure with probability 1-p , are to be performed.
Let X be no of success that occur in the n trials then X is said to be binomial Random variable with parameter (n,p)
Condition of Binomial distribution
\[\frac{n!}{(n-x)!x!}(p^x)(1-p)^{n-x}\]
- Only two outcomes are possible namely success and failure.
- Probability of success p and failure 1-p does not change from trial to trial.
- The trials are statistically independent.
Mean of Binomial distribution= np
variance of Binomial distribution=np(1-p)
Example- 100 dice are thrown . how many are expected to fall 6 . what is the variance of the number 6's?
solution :
Expected value E[X]= np=100*\[\frac{1}{6}\]=16.7
so 17 out of 100 expected to fall 6.
Variance V[x]= np(1-p)= 100*\[\frac{1}{6}\]*\[\frac{5}{6}\]=13.9
Hence variance of the number is 13.9
Boundary conditions
n must be inetger
p must lie in the interval [0,1]
