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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UpStudy AI · Accepted Answer

The optimal strategy is to put only the red ball in one box and all three white balls in the other box—this is answer A.

Here's why:

After shuffling, you randomly choose one of the two boxes with equal probability (½ each). If you choose the box containing only the red ball, you are guaranteed to draw the red ball. If you choose the box with only white balls, you cannot draw the red ball. Therefore, the overall probability is

½ × 1 (red ball if chosen) + ½ × 0 (white balls if chosen) = ½.

Comparing with the other options:
• Option B puts the red ball with one white ball in one box (giving a 50% chance within that box), but then the overall chance becomes ½ × ½ + ½ × 0 = ¼.
• Option C puts the red ball with two white balls (1/3 chance within that box), resulting in ½ × 1/3 + ½ × 0 = 1/6.

Thus, option A gives the highest probability of winning, 50%.
To maximize your chances of drawing the red ball, place the red ball in one box and all three white balls in the other box. This way, you have a 50% chance of selecting the box with the red ball and winning the prize.

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