Arranging balls in two boxes
You must place 50 red balls and 50 blue balls into two boxes, with at least one ball in each box. Then one of the two boxes is chosen with probability , and a ball is drawn uniformly at random from it.
How should you arrange the balls to make a red draw as likely as possible? Give that probability.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Each box counts for half, however many balls it holds. Can one box be a sure win?
Put a single red ball in one box.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Each box carries weight regardless of its size. So make one box a certain win using as few red balls as possible, a single red ball, and put everything else in the other box.
Step 1: the arrangement. Box 1: one red ball. Box 2: 49 red and 50 blue balls.
Step 2: the probability.
Why it is best. A box contributes at most , and it does so only if it holds red balls alone. Once one box is all red, every red ball kept there beyond the first lowers the red share of the other box. Going through all possible splits confirms that is the maximum.
The lesson. Without any arrangement (all balls in one box, or a perfect half split), the probability is . Choosing how to group outcomes can shift the odds a lot when the grouping is chosen before the draw.