Multiple choice

A bag $A $ contains $3$ white and $2$ black balls and another bag $B$ contains $2$ white and $4$ black balls. $ A$ bag and a ball out of it are picked at random. What is the probability that the ball is white?

  1. $\displaystyle \frac{14}{15}$
  2. $\displaystyle \frac{8}{15}$
  3. $\displaystyle \frac{1}{15}$
  4. $\displaystyle \frac{7}{15}$
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D Correct answer
Explanation

P(A) = 1/2, P(B) = 1/2. P(White|A) = 3/5. P(White|B) = 2/6 = 1/3. Total P(White) = P(A)*P(White|A) + P(B)*P(White|B) = (1/2 * 3/5) + (1/2 * 1/3) = 3/10 + 1/6 = (9+5)/30 = 14/30 = 7/15.

AI explanation

By the law of total probability, the probability of selecting a white ball is the sum of the probabilities of choosing bag A and drawing a white ball, plus choosing bag B and drawing a white ball. Since the probability of picking either bag is 1 over 2, the calculation becomes one half multiplied by three fifths, plus one half multiplied by two sixths. Adding these terms together gives three tenths plus one sixth, which equals nine thirtieths plus five thirtieths, resulting in fourteen thirtieths, or seven fifteenths.

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