Question

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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 she 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? Probability = b. Suppose you arrange the balls in the way that maximizes her chances of drawing the black ball. What is her chance of winning? Probability = c. What are the analogous probabilities when there are n balls total (one black, n-1 white) instead of 4 balls total? Minimizing probability = Maximizing probability =

Answer 1

Ans: Let A to the event of choosing an uml and Bc B be the event of choosing black ball. Stice there are 2 ums, P(A) = p (A) = 2 probabilody of choosing Black ball - 8 (B) he know (B) = P(BRA) + P(BRA A = P(BIA). P(A2+ P (B/A) PCA) these will be to siduations of choosong. They are " Drawing a black ball when all balls are is one on and other as empty. Here P (BA) = 7 and P (3/5) = 0 Co PIB) = t (+) + 6( 4) = 3 when uml has I black ball and one has all the 3 whote balls P(B/A) = 1, P (BA) = 0 so P(B) = 1x + ofź) - 12 it when uml has I ballk and I whate where vine has rest of whate balls P(B/A) = P(BIA) = 0

So, P(B) = -3 (b) to (2) - 15 when urnl has I black & white balls and uma has a whote ball I P[B/A) = P(BIA) = 0 So, 8(B) = 1 / 3 (6) + 0 (6) = 1 a) If all the balls are an one on her I chance of wonneng as very less e b) puthing black ball alone in one orn increases her chance of worning with probability 2 c) Smilarly the chance of worming with a masomon probability of can be achieved by taking black ball alone in one um and rest of nt white balls In the other un Minimum probabibay can be obtained when all then balls are taken in one un Suppose all the balls are taken in unch then, Moramum probability = probabilody of getting black ball giren urnix p(selecting omi) - tal- Madani sing probability =