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 is again chosen at random. The probability that P =Q, is

Answer

To solve the problem, we need to find the probability that two randomly chosen subsets \( P \) and \( Q \) of a set \( A \) containing \( n \) elements are equal. Let's break it down step by step.

### Step 1: Determine the total number of subsets of set \( A \)
A set containing \( n \) elements has a total of \( 2^n \) subsets. This includes all possible combinations of the elements, including the empty set and the set itself.

**Hint:** Recall that for any set with \( n \) elements, the number of subsets is given by \( 2^n \).

### Step 2: Calculate the total ways to choose subsets \( P \) and \( Q \)
Since both subsets \( P \) and \( Q \) can be chosen independently from the set \( A \), the total number of ways to choose both subsets is:
\[ 2^n \times 2^n = 4^n \]
This is because for each subset \( P \) there are \( 2^n \) choices for \( Q \).

**Hint:** Remember that the choices for \( P \) and \( Q \) are independent, so you multiply the number of choices.

### Step 3: Determine the number of favorable outcomes where \( P = Q \)
For the subsets \( P \) and \( Q \) to be equal, they must be the same subset. There are \( 2^n \) possible subsets that \( P \) can be, and if \( P \) is chosen, \( Q \) must be that same subset. Thus, there are \( 2^n \) favorable outcomes where \( P = Q \).

**Hint:** Consider that there is only one way to choose \( Q \) once \( P \) has been chosen for each of the \( 2^n \) subsets.

### Step 4: Calculate the probability that \( P = Q \)
The probability \( P \) is equal to \( Q \) is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2^n}{4^n} \]
This simplifies to:
\[ \frac{2^n}{(2^2)^n} = \frac{2^n}{2^{2n}} = \frac{1}{2^{n}} \]

**Hint:** When simplifying fractions with exponents, remember to subtract the exponents in the denominator from those in the numerator.

### Final Answer
The probability that \( P = Q \) is:
\[ \frac{1}{2^n} \]