Multiple choice

One bag contains $3$ white and $2$ black balls, another bag contains $5$ white and $3$ black balls. If a bag is chosen at random and a ball is drawn from it, what is the change that it is white?

  1. $\dfrac{3}{8}$
  2. $\dfrac{49}{80}$
  3. $\dfrac{8}{13}$
  4. $\dfrac{1}{2}$
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B Correct answer
Explanation

P(W) = P(Bag1)P(W|Bag1) + P(Bag2)P(W|Bag2) = (1/2)(3/5) + (1/2)(5/8) = 3/10 + 5/16 = (24+25)/80 = 49/80.

AI explanation

Using the law of total probability, we calculate the chances of drawing a white ball by summing the probabilities of choosing each bag and then drawing a white ball from it. The probability of selecting the first bag is 1 divided by 2, and the chance of a white ball from it is 3 divided by 5. The probability of selecting the second bag is 1 divided by 2, and the chance of a white ball from it is 5 divided by 8. Adding these products gives (1 divided by 2 multiplied by 3 divided by 5) plus (1 divided by 2 multiplied by 5 divided by 8), which equals 3 divided by 10 plus 5 divided by 16, resulting in a final probability of 49 divided by 80.

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