Multiple choice

Two bags contains respectively 3 white and 2 red balls and 2 white and 4 red balls. One ball is drawn at random from the first bag and put it into the second; then a ball is drawn from the second is white is

  1. $\frac{13}{35}$
  2. $\frac{13}{25}$
  3. $\frac{12}{35}$
  4. $\frac{12}{25}$
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A Correct answer
Explanation

Bag 1: 3W, 2R (Total 5). Bag 2: 2W, 4R (Total 6). Case 1: White drawn from B1 (3/5), B2 becomes 3W, 4R (Total 7). Prob(W from B2) = 3/7. Case 2: Red drawn from B1 (2/5), B2 becomes 2W, 5R (Total 7). Prob(W from B2) = 2/7. Total prob = (3/5 * 3/7) + (2/5 * 2/7) = 9/35 + 4/35 = 13/35.

AI explanation

Using the law of total probability, we evaluate the two possible scenarios for transferring a ball from the first bag to the second. The chance of transferring a white ball is 3/5, making the second bag contain 3 white and 4 red balls, which gives a probability of drawing a white ball of 3/7. The chance of transferring a red ball is 2/5, making the second bag contain 2 white and 5 red balls, which gives a probability of drawing a white ball of 2/7. The total probability is (3/5 times 3/7) plus (2/5 times 2/7), which equals 9/35 plus 4/35, resulting in 13/35.

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