I've got this assignment in School and it's supposed to be done by Monday. I am to provide a solution to this math problem.
"You have 50 white and 50 black balls. You are to place them all in two jars, with any number of balls of each color in each jar. Then, you are to pick a jar at random, and from that jar, a ball at random. If it's a white ball, you win, and if it's a black ball, you lose. Basically, the balls should be placed in such a way that the probability of picking a white ball is as large as possible. How many white balls and how many black balls should there be in each jar to maximise the chance to win?"
Common sense and a bit of thinking leads to this conclusion: If I place only one white and no black balls in one jar, and the rest of the balls (99) in the other jar, I will have roughly 75% chance of winning (1/1 + 49/99)/2. However, how do I prove that this is correct without having to try all combinations?
The formula looks something like this:
(P is the probability of picking a white ball, W is the number of white balls in jar #1, and B is the number of black balls in jar #1)
P = ( W/(W+B) + (50-W)/(50-W+50-B) ) / 2
But I can't seem to simplify it into something that really proves that my (very educated) guess is right.
Thanks in advance.
[Edit: annoying emoticons]
"You have 50 white and 50 black balls. You are to place them all in two jars, with any number of balls of each color in each jar. Then, you are to pick a jar at random, and from that jar, a ball at random. If it's a white ball, you win, and if it's a black ball, you lose. Basically, the balls should be placed in such a way that the probability of picking a white ball is as large as possible. How many white balls and how many black balls should there be in each jar to maximise the chance to win?"
Common sense and a bit of thinking leads to this conclusion: If I place only one white and no black balls in one jar, and the rest of the balls (99) in the other jar, I will have roughly 75% chance of winning (1/1 + 49/99)/2. However, how do I prove that this is correct without having to try all combinations?
The formula looks something like this:
(P is the probability of picking a white ball, W is the number of white balls in jar #1, and B is the number of black balls in jar #1)
P = ( W/(W+B) + (50-W)/(50-W+50-B) ) / 2
But I can't seem to simplify it into something that really proves that my (very educated) guess is right.
Thanks in advance.
[Edit: annoying emoticons]