Can someone please help me?

Question

Suppose you have 10 white balls, 10 black balls, and 2 baskets. You may place some of the balls into one basket and the remaining balls into the other basket however you wish. After you are done, your friend Janelle will flip a fair coin to select one of the baskets, then randomly select one ball from it. If you want to maximize the probability that Janelle will select a white ball, how should you distribute the 20 balls into the 2 baskets? And what is the probability Janelle draws a white ball?

Answer 1

This is a kind of tricky question.. If you put all the white balls in basket one, the probability of getting a white ball from basket 1 is 1 and the probability of getting a white ball from basket 2 is zero (all the white balls are in basket 1.).. given the probability of getting basket 1 is 1/2, the probability of getting a white ball is 1/2.. If we split them equally the probability of getting a white ball in each basket is 1/2. and the probability of choosing each basket is 1/2. so the probability of getting a white ball is the prob of choosing basket one times Prob of getting a white ball in basket 1 + prob of choosing basket two times Prob of getting a white ball in basket 1. More formally let X= prob of choosing basket 1 and p1 and p2 be the probabilities of getting a white ball in basket 1 or basket 2. The Probability = Xp1 +(1-X)p1. You should be able to convince your self that if it were not a fair coin (head comes up 55% time) then you would maximize the probability of getting a white ball by putting them all into the basket which is chosen when heads appears on the coin toss.

Hope this helps

Lenny

Answer 2

Hi Roland, thanks for posting.

You have two choices to make: W, the number of white balls in the first basket, and B, the number of black balls in the first basket. The probability of selecting a white ball P(W,B), is 1/2 * W/(W+B) + 1/2 * (10-W)/(20-W-B). I'd be happy to talk you through how to find the maximum value in a lesson.

Answer 3

Put 1 white ball in Basket A, put 9 white balls and 10 black balls in Basket B.

So, if Janelle selects Basket A, P(White) = 1,

and if Janelle selects Basket B, P(White) = 9/19

Since Janelle selects Basket A and Basket B with probability 1/2,

overall probability of drawing White is 1/2 * 1 + 1/2 * 9/19 = 28/38 = 14/19 which is about .74

A little additional work will show that all other arrangements give lower probabilities for drawing a white.