You are given three white balls, one red ball, and two identical boxes. You are asked to distribute the balls in the boxes in any way you like. You then are asked to select a box (after the boxes have been shuffled) and to pick a ball at random from that box. If the ball is red you win a prize. How should you distribute the balls in the boxes to maximize your chances of winning? Justify your reasoning. Choose the correct answer below. A. Put the red ball in one of the boxes and the three white balls in the other box. In this case the probability of the event of getting the red ball is \( \frac{1}{2} \cdot 1+\frac{1}{2} \cdot 0 \), or \( \frac{1}{2} \). B. Put the red ball and one white ball in one of the boxes and the other two white balls in the other box. In this case the probability of the event of getting the red ball is \( \frac{1}{2} \cdot \frac{1}{2}+\frac{1}{2} \cdot 0 \), or \( \frac{1}{4} \). C. Put the red ball and two of the white balls in one of the boxes and the other white ball in the other box. In this case the probability of the event of getting the red ball is \( \frac{1}{2} \cdot \frac{1}{3}+\frac{1}{2} \cdot 0 \), or \( \frac{1}{6} \).
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To maximize your chances of winning the prize by selecting the red ball, the best strategy is option A. By putting the red ball in one box and all three white balls in the other, you have a straightforward calculation for the probability of getting the red ball. When you pick a box at random, you have a 50% chance of selecting the box with the red ball. If you do choose that box, you are guaranteed to pick the red ball, yielding an overall winning probability of \( \frac{1}{2} \). In options B and C, the probabilities of obtaining the red ball diminish significantly because the distribution adds extra uncertainty. For B, with only one white ball in the red ball's box, the chance drops to \( \frac{1}{4} \). For C, since the red ball shares a box with two white balls, the probability sinks to \( \frac{1}{6} \). So, stick with option A and go for that full \( \frac{1}{2} \) chance of winning!
