Probability
Total Probability and Bayes' Theorem
Grade 12
Question:
<p>A card from a pack of 52 cards is lost. From the remaining cards of the pack two cards are drawn and are found to be spades. Find the probability of the missing card to be a spade.</p>
<p>\(\dfrac{11}{50}\)</p>
<p>\(\dfrac{66}{1275}\)</p>
<p>\(\dfrac{11}{1275}\)</p>
<p>\(\dfrac{2}{50}\)</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem by conditioning on whether the lost card is a spade or not. The probability of drawing 2 spades from the remaining 51 cards depends on how many spades are left.
<p><strong>Step 1:</strong> Let S = event that lost card is a spade, and D = event that 2 drawn cards are spades.</p><p>We need P(S|D) using Bayes' theorem: P(S|D) = P(D|S)·P(S) / P(D)</p><p><strong>Step 2:</strong> Calculate P(S) = 13/52 = 1/4 (13 spades in 52 cards)</p><p>Calculate P(S') = 39/52 = 3/4 (lost card is not a spade)</p><p><strong>Step 3:</strong> If lost card is a spade: 12 spades remain in 51 cards<br/>P(D|S) = C(12,2)/C(51,2) = 66/1275 = 22/425</p><p><strong>Step 4:</strong> If lost card is not a spade: 13 spades remain in 51 cards<br/>P(D|S') = C(13,2)/C(51,2) = 78/1275 = 26/425</p><p><strong>Step 5:</strong> P(D) = P(D|S)·P(S) + P(D|S')·P(S')<br/>P(D) = (22/425)·(1/4) + (26/425)·(3/4) = (22 + 78)/(425·4) = 100/1700 = 1/17</p><p><strong>Step 6:</strong> P(S|D) = [(22/425)·(1/4)] / (1/17) = (22/1700) · 17 = 22/100 = <strong>11/50</strong></p><p>∴ Answer: A</p>
Correct Answer: A