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