Recall the game played by two friends named Aaron and Blake. The game is now expanded. Blake puts different things inside each of 5 boxes. Within one box is a money prize of $5, one is a money prize of $2 and within the other three boxes are worthless prizes (scraps of blank paper). To play the game Aaron must pay $2. Next Aaron chooses one box but does not yet open the box. Remember before the game starts only Blake knows where the money prize is located. (a) Calculate the probability that Aaron originally selects the box hiding any money. (b) Under the original game what is the expected net winnings (or loss) if Aaron plays this game. Here net winnings is the payment minus the cost to play. (c) Before letting Aaron open his box, Blake will open one of the other boxes showing Aaron one of the duds. Aaron is then offered the option of switching to a different unopened box or staying with his original box. There are two possible strategies: stay with the initial choice or switch to one of the other remaining unopened box? i. What is Aaron expected winnings if he stays with his original box? ii. What is the probability Aaron wins the best prize if he switches to one of the other boxes? iii. Write the probability probability distribution for the different net winnings assumning Aaron implements the switch strategy. iv. Write the cumulative distribution function for Aaron's net winnings assuming Aaron used the switch strategy. v. Assume Aaron decides to use the switch strategy. In this case what are Aaron's expected net winnings? vi. What is the amount Aaron should pay instead of $2) to make this a perfectly fair game? I. e the expected value for the net winnings is zero.