A is a set conaining 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 is again chosen at random. find the probability that1 P and Q are disjoint sets

hi srinivas

first let us find out the sample space

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 + n C 1 + n C 2 + ............  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 sample space = 2 ⁿ       *    2 ⁿ       = 4 ⁿ   

now no of favourable ways

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_r=0 ⁿC_r (2)^(n-r)=(3)ⁿ

so the probability is (3/4) ⁿ

Approved