Suppose box X has 2 blue and 8 gold balls, and box Y has 8 blue and 2 gold balls, and both Jack and Jill agree on this. One of the two boxes will be picked, with some unknown probability. You must say which box was picked, and if you guess correctly you win $50, otherwise you win nothing. You can purchase one draw from the chosen box for $1. (You will learn the color of the ball before you have say which box was picked.) Suppose Jack believes that box X was chosen with probability 0.8, while Jill believes box X was chosen with probability 0.2. Derive the optimal decisions (buying extra information, choosing a box) for Jack, Jill, and you under the assumption that you will combine Jack's and Jill's assessments of their probabilities that box X was chosen.