Multiple choice

A bag contains $6$ white and $4$ black balls. Another bag contains $4$ white and $6$ black balls. A die is rolled and if it shows a prime number a ball is taken from the 1st bag, otherwise a ball is taken from the second bag. The probability that the ball drawn is white is

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

Probability of prime (2, 3, 5) = 3/6 = 1/2. If prime, pick from bag 1 (6/10 white). If not prime, pick from bag 2 (4/10 white). Total probability = (1/2 * 6/10) + (1/2 * 4/10) = 3/10 + 2/10 = 5/10 = 1/2.

AI explanation

A die has 3 prime numbers, so the probability of choosing the first bag is 3/6 or 1/2, and the probability of choosing the second bag is also 1/2. The probability of drawing a white ball from the first bag is 6/10, and from the second bag is 4/10. By the law of total probability, the combined probability is (1/2 * 6/10) plus (1/2 * 4/10), which equals 1/2.

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