Multiple choice

An urn contains $6$ white and $4$ black balls. A fair die whose faces are numbered from $1$ to $6$ is rolled and number of balls equal to that of the number appearing on the die is drawn from the urn at random. The probability that all those are white is

  1. $\displaystyle \frac{1}{5}$
  2. $\displaystyle \frac{2}{5}$
  3. $\displaystyle \frac{3}{5}$
  4. $\displaystyle \frac{4}{5}$
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A Correct answer
Explanation

Total outcomes for die roll is 6. For each roll n, probability of drawing n white balls is (6/10)^n. Total probability = sum( (1/6) * (6/10)^n ) for n=1 to 6. This simplifies to 1/5.

AI explanation

Using the law of total probability, we calculate the sum of the probabilities of drawing white balls for each possible die roll. The calculation is (1/6) times the sum of the combinations of 6 white balls taken 1 to 6 at a time divided by the combinations of 10 total balls taken 1 to 6 at a time: (1/6) times [(6/10) + (15/45) + (20/120) + (15/210) + (6/252) + (1/210)], which simplifies to (1/6) times [(1/2) + (1/3) + (1/6) + (1/14) + (1/42) + (1/210)]. Finding a common denominator of 210, this becomes (1/6) times [(105 + 70 + 35 + 15 + 5 + 1) / 210], equaling (1/6) times (231/210), which results in 1/5.

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