Optimize red-ball draw probability, prove optimality

Q: Optimize red-ball draw probability, prove optimality

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Question

Two-Box Ball Allocation to Maximize Probability of Drawing Red

Setup

You have 2 boxes and two colors of balls.
In the 100/100 case: 100 red and 100 blue balls.
A box is chosen uniformly at random (probability 1/2 each), then one ball is drawn uniformly from that box.

Tasks

For 100 red and 100 blue balls, how should you distribute the balls between the two boxes to maximize the probability of drawing a red ball? Compute the resulting probability.
Prove optimality (not just intuition).
Generalize: For R red and B blue balls (R,B ≥ 1), characterize the optimal allocation and give the maximal probability as a function of R and B.

Optimize red-ball draw probability, prove optimality

Two-Box Ball Allocation to Maximize Probability of Drawing Red

Setup

Tasks

Solution (Locked)

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