Question

Set S has 4 elements, A and B are subsets of S. The probability thatA and B are not disjoint is
A. \[\dfrac{{175}}{{256}}\]
B. \[\dfrac{{173}}{{256}}\]
C. \[\dfrac{{85}}{{128}}\]
D. \[\dfrac{{45}}{{64}}\]

Answer 1

Hint: The first thing to be done in this question is to find the total number of subsets of S and the total combination of A and B such that A and B will be disjoint sets and then we can finally move on to find the probability.

Complete step by step answer:
So we know that there are a total of \[{2^n}\] number of subsets in a set with n number of elements.
Which means that the total number of subset for the set S is \[{2^4} = 2 \times 2 \times 2 \times 2 = 16\]
Now let us try to find out the total combination of A and B.
A and b can both have 16 subset each which means that the total number of combination of subset becomes \[16 \times 16 = 256\]
As 256 was the total combination of subset then the total disjoint combination of subset will be \[{3^4} = 3 \times 3 \times 3 \times 3 = 81\]
Which means that the probability that A and B are not disjoint is \[\dfrac{{256 - 81}}{{256}} = \dfrac{{175}}{{256}}\]

So, the correct answer is “Option A”.

Note: In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B and two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.