3 Infinity (40 points) Infinitely many people have been turned into statues, and are arranged in two rows with an aisle in between. Rama will walk down the aisle and reanimate (ie., bring back to life) some number of them depending on the outcomes of some coin tosses. Any person reanimated by Rama has an equally good, wonderful life, and remaining inanimate is neutral as opposed to good or bad. Before Rama reanimates anybody this is the arrangement of the statues, o ) @ 3 3 33388 2 3 4 5 6 7 B 9 .. A nickel and a dime will both be tossed. The nickel will be tossed until it lands tails for the first time and the dime will be tossed just once. All the coin tosses are fair and independent of one another. You have two options: (A) If the dime lands heads, Rama will reanimate the first 2 X 2" people in the top row, where n is the number of tosses it took for the nickel to land tails for the first time. If the dime lands tails, then Rama will reanimate the first 2 X 2" people in the bottom row, where n is the number of tosses it took for the nickel to land tails for the first time. (B) Whether the dime lands heads or tails, Rama will reanimate the first 2" people in the top row and also the first 2" statues in the bottom row, where again n is the number of tosses it took for the nickel to land tails for the first time. The outcome on option (A} if the nickel lands heads first and then tails, and the dime lands heads. e 8 8888888 1 2 3 4 5 6 7 & 9 .- The outcome on option (A) if the nickel lands heads first and then tails, and the dime lands tails. w3 333388838 - Botomrow @Y @ @ 1 2 3 4 5 6 7 8 9 - The outcome on option (B) if the nickel lands heads first and then tails. The outcome on option (A) if the nickel lands heads first and then tails, and the dime lands heads. o P @D OO Hdi o s a s a a s 4 5 9 The outcome on option (A} if the nickel lands heads first and then tails, and the dime lands tails. v 3333833843 ~ @ ~ @@ - @ @ 1 2 3 4 5 6 7 8 The outcome on option (B) if the nickel lands heads first and then tails. w9000 884848 w000 d 88848 1 2 3 4 5 & 7 8 9 (i) (5 points) Is it more, less, or equally likely for the first person in the top row to be reanimated on option (A) versus on option (B)? (i) (5 points) For k > 2 it is equally likely for the k person in the top row to be reanimated on option (A) or option (B), and the same goes for the k' person in the bottom row. Explain why. (iii) (10 points) In light of this, what does Ex Ante Pareto say about which is better out of option (A) and option (B)? (iv) (5 points) What does the principle of Equality say about the following three outcomes (represented in the figures above): (a) Exactly the first 8 people in the top row are reanimated. (b) Exactly the first 8 people in the bottom row are reanimated. (c) Exactly the first 4 people in both rows are reanimated. (v) (5 points) How likely is outcome (c) on option (B)? How likely are out- comes (a) and (b) on option (A)7? (vi) (10 points) In light of this, what do the principles of Equality and of Stochastic Equivalence say about which is better out of option (A) and option (B)