<p>Consider the following assignments of probabilities for outcomes of sample space \(S = \{1, 2, 3, 4, 5, 6, 7, 8\}\).</p><table border='1'><tr><th>Number (X)</th><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td></tr><tr><th>Probability P(X)</th><td>0.15</td><td>0.23</td><td>0.12</td><td>0.10</td><td>0.20</td><td>0.08</td><td>0.07</td><td>0.05</td></tr></table><p>Find the probability that <strong>(a)</strong> \(X\) is a prime number.</p>
Step-by-Step Solution
Key Concept: Identify all prime numbers in the sample space S = {1, 2, 3, 4, 5, 6, 7, 8}, then sum their corresponding probabilities from the given probability distribution.
<p><strong>Step 1:</strong> Identify prime numbers in S = {1, 2, 3, 4, 5, 6, 7, 8}</p><p>Prime numbers are those with exactly two distinct positive factors (1 and itself).</p><p>Primes in S: {2, 3, 5, 7} [Note: 1 is NOT prime; 4, 6, 8 are composite]</p><p><strong>Step 2:</strong> Find P(X is prime) by adding probabilities</p><p>P(X ∈ {2, 3, 5, 7}) = P(X=2) + P(X=3) + P(X=5) + P(X=7)</p><p>= 0.23 + 0.12 + 0.20 + 0.07</p><p>= 0.62</p><p><strong>∴ Answer: 0.62</strong></p>
Correct Answer: 0.62