# Explain why the experiment of tossing a coin three times is said to have binomial distribution.

Tool Box     .   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 Solution 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   .

Toolbox:
• In tossing of cans 3 times the number of trial is finite
• Each trial has only 2 possible outcomes
• success or failuer
• Each through "trial"is independent of each other
• Randomvarible X={0,1,2,3,}
P(success)=$$\frac{1}{2}$$
P(failer)=1-$$\frac{1}{2}$$=$$\frac{1}{2}$$
N=3