Cheating dealers have become a serious problem at the mini-Blackjack tables. A miniBlackjack deck has 3 card types (5,10,11) and an honest dealer is equally likely to deal each of the 3 cards. When a player holds 11, cheating dealers deal a 5 with probability 1/4 , 10 with probability 1/2 , and 11 with probability 1/4 . You estimate that 4/5 of the dealers in your casino are honest (H = true) while 1/5 are cheating (H = false). 

a. You see a dealer deal an 11 to a player holding 11. What is the probability that the dealer is cheating? 

The casino has decided to install a camera to observe its dealers. Cheating dealers are observed doing suspicious things on camera (C = true) 4/5 of the time, while honest dealers are observed doing suspicious things 1/4 of the time. 

b. Draw a Bayes net with the variables H (honest dealer), D (card dealt to a player holding 11), and C (suspicious behavior on camera). Write the conditional probability tables.

c. List all conditional independence assertions made by your Bayes net. 

d. What is the probability that a dealer is honest given that he deals a 10 to a player holding 11 and is observed doing something suspicious?

You can either arrest dealers or let them continue working. If you arrest a dealer and he turns out to be cheating, you will earn a $4 bonus. However, if you arrest the dealer and he turns out to be innocent, he will sue you for -$10. Allowing the cheater to continue working will cost you -$2, while allowing an honest dealer to continue working will get you $1. Assume a linear utility function U(x) = x. 

e. You observe a dealer doing something suspicious (C) and also observe that he deals a 10 to a player holding 11. Should you arrest the dealer? 

f. A private investigator approaches you and offers to investigate the dealer from the previous part. If you hire him, he will tell you with 100% certainty whether the dealer is cheating or honest, and you can then make a decision about whether to arrest him or not. How much would you be willing to pay for this information?