Multiple choice

One bag contains 5 white and 3 black balls and an other contains 4 white, 5 red balls. Two balls are drawn from one of them choosing at random.The probability that they are of different colours is

  1. $\displaystyle \frac{15}{56}$
  2. $\displaystyle \frac{5}{18}$
  3. $\displaystyle \frac{275}{504}$
  4. $\displaystyle \frac{275}{624}$
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C Correct answer
Explanation

Probability of choosing bag 1 is 1/2. P(diff colors | bag 1) = (5/8 * 3/7) + (3/8 * 5/7) = 30/56 = 15/28. Probability of choosing bag 2 is 1/2. P(diff colors | bag 2) = (4/9 * 5/8) + (5/9 * 4/8) = 40/72 = 5/9. Total probability = 1/2 * (15/28 + 5/9) = 1/2 * ((135 + 140) / 252) = 275 / 504.

AI explanation

Using the law of total probability, the overall probability is the average of the probabilities of drawing two balls of different colours from each bag. For the first bag, the probability is (5 white * 3 black) / (8C2), which equals 15/28. For the second bag, the probability is (4 white * 5 red) / (9C2), which equals 20/36. The combined probability calculates to (1/2) * (15/28 + 20/36), which equals (1/2) * (135/252 + 140/252), resulting in 275/504.

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