<p>An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well-shuffled pack of 11 cards numbered 2, 3, 4, ..., 12 is picked and the number on the card is noted. What is the probability that the noted number is either 7 or 8?</p>
<p>\(\dfrac{193}{792}\)</p>
<p>\(\dfrac{17}{132}\)</p>
<p>\(\dfrac{91}{792}\)</p>
<p>\(\dfrac{102}{792}\)</p>
Step-by-Step Solution
Key Concept: Use the law of total probability by conditioning on the coin outcome: P(7 or 8) = P(7 or 8|H)·P(H) + P(7 or 8|T)·P(T). Calculate the probability separately for dice and card outcomes, then combine.
<p><strong>Step 1: Find P(7 or 8 | Head) - Dice outcome</strong></p><p>When coin shows head, roll two dice (36 equally likely outcomes).</p><p>Ways to get 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 ways</p><p>Ways to get 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 ways</p><p>P(7 or 8 | H) = (6+5)/36 = 11/36</p><p><strong>Step 2: Find P(7 or 8 | Tail) - Card outcome</strong></p><p>When coin shows tail, pick from 11 cards numbered {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.</p><p>Favorable outcomes: cards numbered 7 and 8 → 2 cards</p><p>P(7 or 8 | T) = 2/11</p><p><strong>Step 3: Apply Law of Total Probability</strong></p><p>P(7 or 8) = P(7 or 8 | H)·P(H) + P(7 or 8 | T)·P(T)</p><p>P(7 or 8) = (11/36)·(1/2) + (2/11)·(1/2)</p><p>P(7 or 8) = (1/2)[11/36 + 2/11]</p><p><strong>Step 4: Find common denominator (LCM of 36 and 11 = 396)</strong></p><p>11/36 = 121/396</p><p>2/11 = 72/396</p><p>P(7 or 8) = (1/2)·(193/396) = <strong>193/792</strong></p><p>∴ Answer: A</p>
Correct Answer: A