You play the following game against your friend. You have 2 urns and 4 balls One of the balls is black and the other 3 are white. You can place the balls in the urns any way that you'd like, including leaving an urn empty. Your friend will choose one urn at random and then draw a ball from that urn. (If he chooses an empty urn, he draws nothing.) She wins if she draws the black ball and loses otherwise.

a. Suppose you arrange the balls in the way that minimizes her chances of drawing the black ball. What is her probability of winning?

b. Suppose you arrange the balls in the way that maximizes her chances of drawing the black ball. What is her probability of winning?

c. What are the analogous probabilities when there are n balls total (one black, n-1 white) instead of 4 balls total?

Part a: The case in such a way that the chances are minimized so the case is where all the four balls are in 1 of the urns the probability of her winning is least as 0.125.

Part b: The case in such a way that the chances are maximized so the case where the black ball is in one of the urns and the remaining 3 white balls in the second urn than, the probability of her winning is maximum as 0.5.

Part c: The minimum and maximum probabilities of winning for n number of balls are such that

when all the n balls are placed in one of the urns the probability of the winning will be least as 1/2n when the black ball is placed in one of the urns and the n-1 white balls are placed in the second urn the probability is maximum, as 0.5

Step-by-step explanation:

Let us suppose there are two urns A and A'. The event of selecting a urn is given as A thus the probability of this is given as

P (A) = P (A') = 0.5

Now the probability of finding the black ball is given as

P (B) = P (B∩A) + P (P (B∩A')

P (B) = (P (B|A) P (A)) + (P (B|A') P (A'))

Now there can be four cases as follows

Case 1: When all the four balls are in urn A and no ball is in urn A'

so

P (B|A) = 0.25 and P (B|A') = 0 So the probability of black ball is given as

P (B) = (0.25*0.5) + (0*0.5)

P (B) = 0.125;

Case 2: When the black ball is in urn A and 3 white balls are in urn A'

so

P (B|A) = 1.0 and P (B|A') = 0 So the probability of black ball is given as

P (B) = (1*0.5) + (0*0.5)

P (B) = 0.5;

Case 3: When there is 1 black ball and 1 white ball in urn A and 2 white balls are in urn A'

so

P (B|A) = 0.5 and P (B|A') = 0 So the probability of black ball is given as

P (B) = (0.5*0.5) + (0*0.5)

P (B) = 0.25;

Case 4: When there is 1 black ball and 2 white balls in urn A and 1 white ball are in urn A'

so

P (B|A) = 0.33 and P (B|A') = 0 So the probability of black ball is given as

P (B) = (0.33*0.5) + (0*0.5)

P (B) = 0.165;

Part a:

As it says the case in such a way that the chances are minimized so the case is case 1 where all the four balls are in 1 of the urns the probability of her winning is least as 0.125.

Part b:

As it says the case in such a way that the chances are maximized so the case is case 2 where the black ball is in one of the urns and the remaining 3 white balls in the second urn than, the probability of her winning is maximum as 0.5.

Part c:

The minimum and maximum probabilities of winning for n number of balls are such that

when all the n balls are placed in one of the urns the probability of the winning will be least given as

P (B|A) = 1/n and P (B|A') = 0 So the probability of black ball is given as

P (B) = (1/n*1/2) + (0*0.5)

P (B) = 1/2n;

when the black ball is placed in one of the urns and the n-1 white balls are placed in the second urn the probability is maximum, equal to calculated above and is given as

P (B|A) = 1/1 and P (B|A') = 0 So the probability of black ball is given as

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

P (B) = 0.5;