Multiple choice

A sample of size $4$ is drawn with replacement be the first part of the problem and without replacement be the second part of the problem, then from an urn containing $12$ balls, of which $8$ are white, what is the conditional probability that the ball drawn on the third draw was white, given that the sample contains $3$ white balls ?

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

Given that exactly 3 of the 4 positions are white, symmetry makes each position equally likely to be one of the white positions. Thus, the probability that the third draw is white is 3 out of 4. This remains true for both sampling with replacement and sampling without replacement under the stated condition.

AI explanation

Using the definition of conditional probability, we must find the probability that the third draw is white given that exactly 3 white balls were drawn in the 4 trials. In a random sample of 4 balls containing exactly 3 white balls, any specific draw, including the third one, has an equal chance of being a white ball. Therefore, the conditional probability is the ratio of 3 white balls to the 4 total draws, resulting in three fourths.

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