Multiple choice

An urn contains 5 white and 7 black balls. A second urn contains 7 white and 8 black balls. One ball is transfered from the 1st urn to the 2nd urn without noticing its colour. A ball is now drawn at random from the 2nd urn. The probability that it is white is

  1. $\displaystyle \frac{8}{92}$
  2. $\displaystyle \frac{89}{192}$
  3. $\displaystyle \frac{9}{92}$
  4. $\displaystyle \frac{98}{192}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Case 1: White ball transferred (5/12). Urn 2 now has 8 white, 8 black. P(W) = 8/16 = 1/2. Case 2: Black ball transferred (7/12). Urn 2 now has 7 white, 9 black. P(W) = 7/16. Total P(W) = (5/12 * 1/2) + (7/12 * 7/16) = 5/24 + 49/192 = 40/192 + 49/192 = 89/192.

AI explanation

Using the law of total probability, the chance of transferring a white ball is 5/12 and a black ball is 7/12. If a white ball transfers, the second urn has 8 white balls out of 16, giving a probability of 1/2. If a black ball transfers, the second urn has 7 white balls out of 16. The total probability of drawing a white ball is (5/12 * 1/2) plus (7/12 * 7/16), which equals 89/192.

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