Multiple choice

A fair coin is tossed if the result is a head, a pairof fair dice is rolled and the score is noted. If theresult is a tail, a card from a well shuffled pack ofeleven numbered 1,2,3,.......11 is picked and thenumber on the card is noted. Then the probability that the number noted is either 7 or 8 is

  1. $\displaystyle \frac{45}{512}$
  2. $\displaystyle \frac{193}{792}$
  3. $\displaystyle \frac{193}{1024}$
  4. $\displaystyle \frac{243}{792}$
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B Correct answer
Explanation

P(Head) = 1/2. If Head, roll two dice (36 outcomes). Sum 7 or 8: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) [6] and (2,6),(3,5),(4,4),(5,3),(6,2) [5]. Total 11/36. P(Tail) = 1/2. If Tail, pick from 11 cards. 7 or 8: 2/11. Total P = (1/2 * 11/36) + (1/2 * 2/11) = 11/72 + 1/11 = (121 + 72) / 792 = 193/792.

AI explanation

Using the law of total probability, the chance of noting a 7 or 8 is the sum of the probabilities of getting a 7 or 8 from the dice and getting a 7 or 8 from the cards, each multiplied by the chance of their respective pathways. For the dice, there are 11 favorable outcomes out of 36 to roll a sum of 7 or 8, giving a probability of 11/36, which is multiplied by 1/2 to get 11/72. For the cards, there are 2 favorable cards out of 11, giving a probability of 2/11, which is multiplied by 1/2 to get 1/11. Adding these probabilities, 11/72 plus 1/11, gives 121/792 plus 72/792, resulting in 193/792.

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