Binomial Theorem Probability

Introduction to Binomial theorem probability:-

In a binomial theorem probability, we deal with two outcomes.  They are called 'success' and  'falure'.  These are two mutually
disjoint outcomes.
We denote success by the symbol p and the failure by the symbol q.  Obviously p+q=1
Formula  for binomial distribution is P[X=x] = {ncx.pxqn-x}  x= 0,1,2 ....... n
The two independent constants n and p are called the parameters of the distribution
Mean = np
Variance = npq
Standard deviation =vnpq

I like to share this Binomial Experiment with you all through my article.

Problems on Binomial Theorem Probability.

Problem 1:- Eight coins are tosses simultaneously.  Find the probability of getting atleast 6 heads.
Solution:-
Number of trials n = 8
Probability of getting a head is 1/2   That is p= 1/2  that mean q = 1/2
Formula  p(x) = P(X-x) = ncxpx. qn-x and x= 0,1,2 ..... n
= 8cx (0.5)x.(0.5)n-x
= 8cx(0.5)8
Probability of getting at-least 6 heads is P(X=6) = P(X=6) + P(X=7)+P(X=8)
= p(6) +p(7)+p(8)
= (0.5)8[ 8c6 + 8c7 + 8c8}
= (0.5)8[ 8c2 + 8c1  + 1]   since 8c6 = 8c2 and  8c7=8c1
= (0.5)8[ 28 +8+1]
=  1(37)           37
----           =  -----    or   0.14 
256              256
The answer is 0.14

Problem 2 :  Given n = 6 and 9P(X=4)= P(X = 2)  ,  find p

Formula  for probability of randon variable is  P(X=x) = ncx.px.qn-x   x = 0,1,2,3........n
Here n= 6  q = p-1   and P(X=x) = 6cx pxq6-x; x=0,1,2 .....6
x=4 and x=2   hence P(X=4) = 6c4p4q2= 6c2p4q2  (since 6c4 = 6c2)
P(X=2) = 6c2p2q4
It is given 9P(X=4) = P(X=2)
hence      9.6c2p4q2 = 6c2p2q4 =   9 p2 =q2  

Let us take positive square root of both sides.
Then we get 3p= q  which can be written as 3p = 1-p
Transposing p to the other side we get        4p = 1    or  p = 1/4 = 0.25
Hence p =¼  or 0.25 

Algebra is widely used in day to day activities watch out for my forthcoming posts on how do you write an algebraic expression and algebra 2 homework solver. I am sure they will be helpful.           
                                              
Practice Problems on Binomial Theorem Probability:-


1.In a Binomial disribution the mean is 12 and standard deviation is 2,  Find n and p ( Answer  n=18,  p=2/3)

2.Ten coins are tossed simultaneously.Find the probability of getting
a) atleast 7 heads                                                                                                                     
b) exactly 7 heads                                                                                                                    
c) atmost 7 heads     
Answer  (a) 11/64     (b) 15/128   (c)=121/28     

3.A pair of dice is thrown 4 times.  Getting a doublet is considered a success, find the
probability of 2 success.   (answr 25/216)   

4) For a binomial distribution, mean = 7 and variance is 16.  Is this possible?  ( Answer  impossible)

5) Find the mean of a binomial distribution where  n = 10 and p = 3/5  (Answer   6)