Multiple choice

Three bags contain 3 red, 7 black; 8 red, 2 black, and 4 red & 6 black balls respectively. One of the bags is selected at random and a ball is drawn from it. If the ball drawn is red, find the probability that it is drawn from the third bag.

  1. 1/15

  2. 2/15

  3. 3/15

  4. 4/15
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D Correct answer
Explanation

Using Bayes' Theorem: P(Bag3|Red) = P(Red|Bag3)P(Bag3) / [P(Red|Bag1)P(Bag1) + P(Red|Bag2)P(Bag2) + P(Red|Bag3)P(Bag3)]. Probabilities: 3/10, 8/10, 4/10. Calculation: (4/10 * 1/3) / ((3/10 + 8/10 + 4/10) * 1/3) = 4/15.

AI explanation

Using Bayes' theorem, the probability is P(Red from Bag 3) divided by the total probability of drawing a red ball from any bag. The prior probability of selecting any bag is 1/3, so the total probability of drawing a red ball is (1/3)(3/10) + (1/3)(8/10) + (1/3)(4/10) = 15/30 = 1/2. The probability that it was drawn from the third bag is [(1/3)(4/10)] / (1/2) = (4/30) / (1/2) = 4/15.

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