Question

Let X be a set containing n elements. Two subsets A and B of X are chosen at random. Find the probability that $A \cup B = X$.
(a). Required probability $P(E) = {\left( {\dfrac{1}{4}} \right)^n}$
(b). Required probability $P(E) = {\left( {\dfrac{3}{4}} \right)^n}$
(c). Required probability $P(E) = {\left( {\dfrac{1}{2}} \right)^n}$
(d). Required probability $P(E) = {\left( {\dfrac{5}{8}} \right)^n}$

Answer 1

Hint: Determine the total number of ways of choosing two subset A and B of X. Then, find the number of ways of choosing A and B such that $A \cup B = X$. Then, find the probability.

Complete step by step answer:
Probability is a measure of the likelihood of an event to occur. It is calculated as the ratio of the number of favourable outcomes to the total number of outcomes.
Hence, the formula for probability is as follows:
$P(E) = \dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{Total number of outcomes }}}}.............(1)$
It is given that the set X contains n elements. Let the elements be ${a_1}$, ${a_2}$,. . . . . and ${a_n}$.
X = { ${a_1}$, ${a_2}$, ${a_3}$, . . . , ${a_n}$ }
Each element has four options. They can either belong to both the sets A and B or they can belong to set A and not belong to set B or they can belong to set B and not belong to set A and they can not belong to both the sets A and B.
Hence, the total number of ways of choosing two subsets of X is given as follows.
$N(S) = 4.4.4.4.......4{\text{ }}(n{\text{ }}times)$
$N(S) = {4^n}................(2)$
For the condition, $A \cup B = X$ to be true, all elements should belong to either of the sets A and B leaving them three options out of the four mentioned above.
Hence, the total number of ways of choosing A and B, such that $A \cup B = X$ is given as follows:
$N(E) = 3.3.3.3.......3{\text{ }}(n{\text{ }}times)$
$N(E) = {3^n}..............(3)$
From equations (1), (2) and (3), we have the following:
$P(E) = \dfrac{{N(E)}}{{N(S)}}$
$P(E) = \dfrac{{{3^n}}}{{{4^n}}}$
$P(E) = {\left( {\dfrac{3}{4}} \right)^n}$
Hence, the correct answer is option (b).

Note: To find the number of ways of choosing sets A and B, consider the options available for each element rather than trying to find the sets A and B itself. And that will help to stick to what is asked in the question and avoid needlesS steps.