Question

A is a set containing 'n' elements.A subset P of A is chosen at random.The set A is reconstructed by replacing the elements of P. A subset Q of A is chosen again at random. Find the probability that
(a)P and Q have same number of elements.
(b)The number of elements in P is more than the number of elements in Q.
(c)The number of elements in P is just one more than the number of elements in Q.
(d)Q is a subset of P.
(e)P union Q contains exactly r elements(1<=r<=n).

Answer 1

Thank you for asking this question. Here is your answer:

First we will find this out:

S.S = no. of ways in which we can form set A and no of ways in which we can form set B

It is 2 n  in both the cases ( nC0 + nC 1 + nC2 + ............  nCn )

nC0 when the subset is null set

nC 1 when the subset contains 1 element it goes on when the subset contains all the  elements of the superset .

So the sample space = 2 n x 2 n      

= 4 n    

So the favorable ways are:

When P  subset  contains no element  and Q subset contains n elements

P subset contains 1 element and  Q subsets contains n - 1 elements  

P subset contains r elements and Q subset contains n - r elements

P subset contains n elements and  Q subset contains no element

∑^n base r=0 ^ nC base r (2)^(n-r)=(3)^n

So the probability is (3/4)^ n

If there is any confusion please leave a comment below.