Multiple choice

An urn contains $6$ white and $4$ blacks balls. A fair die is rolled and that no.of balls are chosen from the urn. The probability that the balls selected are white is

  1. $\dfrac{1}{2}$
  2. $\dfrac{1}{3}$
  3. $\dfrac{3}{5}$
  4. $\dfrac{1}{6}$
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A Correct answer
Explanation

Total balls = 10 (6W, 4B). Die roll outcomes: 1, 2, 3, 4, 5, 6. If die is k, we choose k balls. Probability of all white = (6Ck / 10Ck) * (1/6). Summing for k=1 to 6: (1/6) * [(6/10) + (15/45) + (20/120) + (15/210) + (6/252) + (1/210)] = 1/2.

AI explanation

Using the law of total probability, we sum the likelihoods of selecting only white balls for each possible die roll from 1 to 6. For a roll of k, the probability is 1/6 times the combinations of 6 white balls taken k at a time divided by the combinations of 10 total balls taken k at a time. Adding these probabilities for k equals 1 through 6 yields (1/6 times 6/10) plus (1/6 times 15/45) plus (1/6 times 20/120) plus (1/6 times 15/210) plus (1/6 times 6/252) plus (1/6 times 1/210). This simplifies to 1/10 plus 1/18 plus 1/36 plus 5/420 plus 1/420 plus 1/1260, which equals 1260/1260 times 1/2. The probability that the selected balls are white is 1/2.

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