I am new here but I have a question that I would like to ask. If any body is in the know, kindly assist. The problem is from Berger (1985) statistical decision theory and Bayesian Analysis Exercise No. 39 under Topic 4. The problem goes like this:
Consider two boxes A and B each of which contains both red balls and green balls. It is known that, in one of the boxes, $1/2$ of the balls are red and $1/2$ are green and that in the other box, $1/4$ of the balls are red and $3/4$ are green. Let the box in which $1/2$ are red to be denoted box W and suppose $P(W=A)=\xi$ and $P(W=B)=1-\xi$. Suppose that the statistician may select one ball at random from either box A or box B and that, after observing its color, he must decide whether $W=A$ or $W=B$. Prove that if $1/2<\xi<2/3$, then in order to maximize the probability of making a correct decision, he should select the ball from box B. Prove also that if $2/3<\xi<1$, then it does not matter from which box the ball is selected.
I tried my best to sole the problem by obtaining the probabilities of selecting the balls from either of the boxes and thereafter obtain the utilities. My intention is to graph the utilities of selecting the balls from either of the boxes with the given conditions and then proceed and compare the graphs. However, I got stuck when I noticed that my utility functions are constants. Could anybody check my solution and let me know where I might be going wrong. I am wondering whether I have a mistake in the calculation of the utility. Thanks in advance for your assistance. my solution attemptContinuation of my solution attemptcontinuation of my solution attempt
