Black gets more priority
You have 50 white balls, 50 black balls and 2 bags. You want to arrange the balls into these bags in such a way as to maximize the probability that picking a bag at random and then picking a ball at random results in a black ball.
What is the maximum probability?
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2 solutions
@Akshay Ginodia I made it clear that the bag was picked at random. Previously, a possible interpretation was that we could pick the bag we wanted, which would lead to an answer of 1.
Denote x is the number of black balls on bag 1, y is the total number of balls in bag2. It is maximum of the expression: M A X ( 2 1 × [ y x + 1 0 0 − y 5 0 − x ] = 9 9 7 4 )
How do you know that the maximum of the expression is 9 9 7 4 ?
anyway where is that formula come from?
well it is closed to 3/4...../
I think the formula should be Max(0.5(x/100-y + (50-x)/y )) .... On which condition did you maximise the equation??
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This expression is the same, and it gets the same result.
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But ... On which condition did he maximise the expression?? It should not be x+y = 50
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@Sandeep Sunnapu – Condition is: x ≤ y , x ≤ 5 0
Just put one black ball on its own in one bag and then the remaining 99 balls in the second bag. Then if the first bag is chosen, (with probability 1/2), the probability of choosing a black ball is 1. If the second bag is chosen, (with probability 1/2), then the probability of choosing a black ball is 49/99. This gives us an overall probability of
(1/2) * 1 + (1/2) * (49/99) = 7 4 / 9 9 .