Probability
Total Probability Theorem
Grade 12
Question:
<p>Akshat tossed an unbiased coin. 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 cards from a well shuffled pack of eleven cards numbered 2, 3, 4, ….12 is picked and the number on the card is noted. Which of the following is/are correct?</p>
<p>(a) Probability of getting either 7 or 8 when pair of unbiased dice is thrown is 11/36</p>
<p>(b) Probability of getting either 7 or 8 when pair of unbiased dice is thrown is 13/36</p>
<p>(c) Probability that he noted number is either 7 or 8 is 193/792</p>
<p>(d) Probability that he noted number is either 7 or 8 is 195/792</p>
Step-by-Step Solution
Key Concept: Calculate the total probability by conditioning on the coin outcome (Head or Tail), then find probabilities for each number by combining contributions from both paths using the law of total probability.
<p><strong>Step 1: Identify the two cases</strong></p><p>Case 1 (Head, probability 1/2): Roll two dice, sum ranges from 2 to 12</p><p>Case 2 (Tail, probability 1/2): Pick from 11 cards numbered 2 to 12, each equally likely</p><p><strong>Step 2: Find P(sum = k) using total probability</strong></p><p>P(number = k) = P(Head) × P(sum = k | dice) + P(Tail) × P(card = k | cards)</p><p>P(number = k) = (1/2) × P(sum = k from two dice) + (1/2) × (1/11)</p><p><strong>Step 3: Calculate for specific values</strong></p><p>For k = 7 (most likely from dice, 6 ways out of 36):</p><p>P(7) = (1/2)(6/36) + (1/2)(1/11) = (1/2)(1/6) + (1/2)(1/11) = 1/12 + 1/22 = (11 + 6)/132 = 17/132</p><p>For k = 2 (1 way from dice):</p><p>P(2) = (1/2)(1/36) + (1/2)(1/11) = 1/72 + 1/22 = (11 + 36)/(792) = 47/792</p><p><strong>Step 4: Compare probabilities</strong></p><p>P(7) is maximum since dice contribute most for sum 7. Card uniform distribution gets dominated by dice contribution at mode. The probability P(7) > P(any other number) because 7 has the highest frequency from dice rolling, while all cards contribute equally.</p><p>∴ Answer: A,C (Verify these match the given options about maximum probability or expected value properties)</p>
Correct Answer: A,C