. In a random experiment and an event A associated with it , if the experiment results in event A , it is a sucess denoted by S and P( S) = p and the event does not occur it results in a failure denoted by F and P(F) = q . then p+q = 1. the experiment is repeated n number of times .
. Let X denote the random variable which associates every outcome to the number of success in it.Then X assumes values 0,1,2,......,n such that
P(X=r) =nCr p^r q^n-r r = 0,1,2,....n
. The random variable X and the probability distribution function P(X=r) = nCr p^r q^n-r , r = 0,1,2,....n is said to follow Bianomial Distribution
Step 1. If a coin is tossed , there are only two possible outcomes , head with P (H) =1/2 and tail P ( T) =1/2 . The experiment is repeated 3 times , the sucess can be a head or a tail.hence P(S) = p = 1/2 and P(F) = q and p+q = 1.
X a random variable can take on values 0,1 , 2, 3
Step 2. P(X=0) =P(FFF) = nC0 p^0 q^3
P(X=1) = P(FFS,SFF,FSF) =nC1 p^1 q^2
P(X=2) = P(SSF,SFS,FSS) =nC2 p^2 q^1
P(X=3) = P (SSS) = nC3 p^3 q^0
and P(X=0) + P(X=1) +P(X=2) +P(X=3) = ( p+q)^n = 1^n =1
Step 3. Hence tossing of a coin 3 times is a Bianomial distribution