Guessing what box. Consider a game as in Examples 1 and 2, where 1 pick one of the three boxes, then you guess which box 1 picked after seeing the color of a ball drawn at random from the box. Then you learn whether your guess was right or wrong Suppose we play the game over and over, replacing the ball drawn and maxing up the balls between plays. Your objective is to guess the box correctly as often as possible.
a) Suppose you know that I pick a box each time at random (probability 1/3 for each box). And suppose you adopt the strategy of guessing the box with highest posterior probability given the observed color, as described in Example 1, in case the observed color is white. About what proportion of the time do you expect to be right over the long run?
b) Could you do any better by another guessing strategy? Explain
c) Suppose you knew I was either randomizing with probabilities (1/2, 1/4, ¼) instead of (1/3, 1/3, 1/3). Now how would your strategy perform over the long run
d) Suppose you knew I was either randomizing with probabilities (1/3, 1/3, 1/3), or with probabilities (1/2, 1/4, ¼). How could you learn which I was doing? How should you respond, and how would your response perform over the long run?