<p>A card from a pack of 52 cards is lost. From the remaining cards, two cards are drawn and are found to be spades. Find the probability that missing card is also a spade.</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem: P(lost card is spade | 2 spades drawn) = P(2 spades drawn | lost card is spade) × P(lost card is spade) / P(2 spades drawn). The probability of drawing 2 spades depends on whether the lost card was a spade.
<p><strong>Step 1:</strong> Define events: Let S = 'lost card is spade', E = 'two drawn cards are spades'</p><p><strong>Step 2:</strong> Calculate P(E|S): If lost card is spade, 12 spades remain in 51 cards. P(drawing 2 spades) = C(12,2)/C(51,2) = 66/1275</p><p><strong>Step 3:</strong> Calculate P(E|S'): If lost card is not spade, 13 spades remain in 51 cards. P(drawing 2 spades) = C(13,2)/C(51,2) = 78/1275</p><p><strong>Step 4:</strong> Apply Bayes' theorem: P(S|E) = P(E|S)·P(S) / [P(E|S)·P(S) + P(E|S')·P(S')]</p><p><strong>Step 5:</strong> P(S) = 13/52 = 1/4, P(S') = 39/52 = 3/4</p><p><strong>Step 6:</strong> P(S|E) = (66/1275 × 1/4) / [(66/1275 × 1/4) + (78/1275 × 3/4)]</p><p><strong>Step 7:</strong> = 66 / (66 + 234) = 66/300 = 11/50</p><p>∴ Answer: <strong>11/50</strong></p>
Correct Answer: 11/50