Let A be a set consisting of n elements. The probability of selecting two subsets P and Q of set A such that `Q= overline(P)`, is

Options

A. `(1)/(2)`
B. `(1)/(2^(n) - 1)`
C. `(1)/(2^(n))`
D. `(1)/(3^(n))`

Answer

To solve the problem, we need to find the probability of selecting two subsets \( P \) and \( Q \) of a set \( A \) such that \( Q = \overline{P} \) (the complement of \( P \)). Let's go through the steps to derive the solution.

### Step 1: Understand the total number of subsets
A set \( A \) with \( n \) elements has a total of \( 2^n \) subsets. This is because for each element in the set, we have two choices: either to include it in a subset or not.

**Hint:** Remember that the number of subsets of a set with \( n \) elements is \( 2^n \).

### Step 2: Define the subsets \( P \) and \( Q \)
Let \( P \) be any subset of \( A \). The complement of \( P \), denoted as \( \overline{P} \), consists of all elements in \( A \) that are not in \( P \). Thus, if we select a subset \( P \), the subset \( Q \) is automatically determined as \( Q = \overline{P} \).

**Hint:** The complement of a subset contains all the elements of the original set that are not in that subset.

### Step 3: Count the favorable outcomes
For every subset \( P \) that we choose, there is exactly one corresponding subset \( Q \) (which is \( \overline{P} \)). Therefore, the number of favorable outcomes (where \( Q = \overline{P} \)) is equal to the total number of subsets, which is \( 2^n \).

**Hint:** Each subset \( P \) has a unique complement \( Q \).

### Step 4: Calculate the total outcomes
When selecting two subsets \( P \) and \( Q \) independently from the set \( A \), the total number of ways to choose any two subsets is \( 2^n \times 2^n = 2^{2n} \). This is because we can independently choose any subset \( P \) and any subset \( Q \).

**Hint:** The total number of combinations of two independent choices is the product of the number of choices for each.

### Step 5: Calculate the probability
The probability \( P \) of selecting two subsets \( P \) and \( Q \) such that \( Q = \overline{P} \) is given by the ratio of the number of favorable outcomes to the total outcomes:

\[ P(Q = \overline{P}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2^n}{2^{2n}} = \frac{1}{2^n} \]

### Final Answer
Thus, the probability of selecting two subsets \( P \) and \( Q \) such that \( Q = \overline{P} \) is \( \frac{1}{2^n} \).

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