<p>A set \(S\) contains 7 elements. A non-empty subset \(A\) of \(S\) and an element \(x\) of \(S\) are chosen at random. Then the probability that \(x \in A\) is:</p>
<p>\(\dfrac{1}{2}\)</p>
<p>\(\dfrac{64}{127}\)</p>
<p>\(\dfrac{63}{128}\)</p>
<p>\(\dfrac{31}{128}\)</p>
Step-by-Step Solution
Key Concept: For a randomly chosen non-empty subset A and random element x, use conditional probability: count pairs (A,x) where x∈A divided by total pairs (non-empty A, any x). Alternatively, recognize that by symmetry, each element has equal probability of being in a randomly chosen non-empty subset.
<p><strong>Step 1:</strong> Identify the sample space. We choose a non-empty subset A from S and an element x from S. Total number of ways = (2^7 - 1) × 7 = 127 × 7 = 889.</p><p><strong>Step 2:</strong> Count favorable outcomes where x ∈ A. For a fixed element x ∈ S, we need non-empty subsets A that contain x. The number of subsets of S containing x = 2^6 = 64 (we must include x, and choose freely from remaining 6 elements). Since x can be any of 7 elements, total favorable pairs = 7 × 2^6 = 7 × 64 = 448.</p><p><strong>Step 3:</strong> Calculate probability: P(x ∈ A) = 448/889 = (7 × 2^6)/(127 × 7) = 2^6/127 = 64/127.</p><p><strong>Alternative (Symmetry):</strong> By symmetry, each element of S has equal probability of being in a random non-empty subset. For element x, P(x ∈ A) = (# non-empty subsets containing x)/(# non-empty subsets) = 2^6/(2^7 - 1) = 64/127.</p><p>∴ Answer: B</p>
Correct Answer: B