Puzzle | 50 red marbles and 50 blue marbles

Give two boxes B1 and B2 one have 50 red marbles and other have 50 blue marbles. A ball is selected randomly from any of the box and the task is to maximize the probability of selecting a red ball, by reshuffling marbles in both the boxes.

Solution:
Let P(R) be the probability of picking a red marble.

P(R) = P(B1) * P(B1 | J1) + P(B2) * P(B2 | J2)

Here, P(B1) and P(B2) refers to selecting B1 and B2 and the probability of selecting each box is \frac{1}{2} J1 and J2 refers to number of total balls in B1 and B2 respectively.



If we do not reshuffle any balls. Then

P(R) = ((1 / 2) * 1) + ((1 / 2) * 0) = 0.5

But, If we decrease the number of red balls in box B1 and increase the number of red balls in box B2 then the probability of getting a red ball will be maximized.
Therefore, let us take 49 red marbles from B1 to B2 then there will be 1 red ball in B1 and 99 balls in B2 out of which 49 are red and 50 of them are blue in the second jar.
Then

P (R) = ((1 / 2) * (1 / 1)) + ((1 / 2) * (49 / 99)) = 0.747474

Hence,

the maximum probability of choosing a red ball is 0.747474

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