Probability
Random Variables & Distributions
Grade 12
Question:
<p>Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let <i>X</i> denote the random variable of number of aces obtained in the two drawn cards. Then \(P(X = 1) + P(X = 2)\) equals:</p>
<p>\(\dfrac{49}{169}\)</p>
<p>\(\dfrac{52}{169}\)</p>
<p>\(\dfrac{24}{169}\)</p>
<p>\(\dfrac{25}{169}\)</p>
Step-by-Step Solution
Key Concept: With replacement, each draw is independent with P(ace) = 4/52 = 1/13. X=1 means exactly one ace (first ace or second ace), and X=2 means both aces. Use binomial probability for exactly k successes in n trials.
<p><strong>Step 1: Identify probabilities</strong></p><p>P(ace) = 4/52 = 1/13</p><p>P(non-ace) = 48/52 = 12/13</p><p><strong>Step 2: Calculate P(X=1) - exactly one ace</strong></p><p>P(X=1) = P(ace on 1st, non-ace on 2nd) + P(non-ace on 1st, ace on 2nd)</p><p>P(X=1) = (1/13)(12/13) + (12/13)(1/13) = 24/169</p><p><strong>Step 3: Calculate P(X=2) - both aces</strong></p><p>P(X=2) = P(ace on 1st AND ace on 2nd)</p><p>P(X=2) = (1/13)(1/13) = 1/169</p><p><strong>Step 4: Add the probabilities</strong></p><p>P(X=1) + P(X=2) = 24/169 + 1/169 = 25/169 = (5/13)²</p><p>∴ Answer: <strong>25/169</strong></p>
Correct Answer: D