Multiple choice

A box contains $5$ red and $4$ white marbles. Two marbles are drawn successively from the box without replacement and the second drawn marble drawn is found to be white. Probability that the first marble is also while is?

  1. $\displaystyle\frac{3}{8}$
  2. $\displaystyle\frac{1}{2}$
  3. $\displaystyle\frac{1}{3}$
  4. $\displaystyle\frac{1}{4}$
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A Correct answer
Explanation

Given that the second marble is white, there are 8 possible remaining first-marble positions in the conditional sample space, of which 3 correspond to a white first marble. Thus, the probability is 3/8.

AI explanation

We apply Bayes' theorem to find the conditional probability that the first marble is white given that the second marble drawn is white. The probability of drawing a white marble second is 4/9, since the initial proportion of white marbles is maintained. The probability of drawing a white marble first and then a white marble second is 4/9 multiplied by 3/8, which equals 1/6. Dividing this joint probability by the total probability of the second marble being white yields 1/6 divided by 4/9, giving the result 3/8.

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