How can 20 balls, 10 white and 10 black, be put into two urns so as to maximize the probability of drawing a white ball if an urn is selected at random and a ball is drawn at random from it?
Intuitively i know the right answer: put 1 white, 0 blacks in one urn and 9 white and 10 blacks in the other, but i just want to arrive at it with more mathematical arguments.
Here's my attempt:
Define the events:
W:= the ball taken is white
B:=the ball taken is black
$U_1$:=the ball taken is from urn 1
$U_2$:= the ball taken is from urn 2.
We are interested in W event, which can be written as the following disjoint union:
$W= WU_1\cup WU_2 $.
Supposing that we put $u$ balls in the first urn, and that from these $w$ are white, we have the following distribuition:
$U_1:$ $u$ balls, $w$ white and $u-w$ black.
$U_2:$ $20-u$ balls, $10-w$ white and $20-10+w$ black.
I'm actually having trouble in order to maximize it. Can someone help?