Probability
Classical Probability
Grade 12
Question:
<p>Let \(S = \{a, b, c, d, e, f, g\}\) and \(x \in S\). A non-empty subset \(A\) of \(S\) is chosen at random. What is the probability that \(x \in A\)?</p>
<p>\(\frac{63}{127}\)</p>
<p>\(\frac{64}{127}\)</p>
<p>\(\frac{1}{2}\)</p>
<p>\(\frac{32}{127}\)</p>
Step-by-Step Solution
Key Concept: For any fixed element x in S, exactly half of all non-empty subsets contain x. This is because subsets come in pairs: for each subset not containing x, there's a corresponding subset with x added, and vice versa.
<p><strong>Step 1:</strong> Count total non-empty subsets of S.</p><p>Total subsets of S = 2^7 = 128. Non-empty subsets = 128 - 1 = 127.</p><p><strong>Step 2:</strong> Count non-empty subsets containing x.</p><p>If x must be in subset A, we only choose from the remaining 6 elements {S \ {x}}. These 6 elements can form any subset (including empty), giving 2^6 = 64 subsets containing x. All 64 are non-empty since x is already in them.</p><p><strong>Step 3:</strong> Calculate probability.</p><p>P(x ∈ A) = 64/127</p><p>∴ Answer: B</p>
Correct Answer: B