Problem 1. (St.Petersburg Paradox) Consider a coin-tossing game where a player flips a fair coin until a head appears for the first time. The payout for the player is determined by the number of tosses needed to achieve the first head. Specifically, the payout is $2 if a head appears on the first toss, $4 if on the second, and in general, the player wins 2^(k) dollars if the first head appears on the k-th toss. Each player must also pay a fixed entry fee of M dollars to participate.
(a) Calculate the expected value of the player's winnings from this game.
(b) Given an entry fee of M=30, would you choose to participate in this game? Explain your reasoning.
(c) If the entry fee is increased to M=1000, does your decision to participate change? Explain your reasoning.
(d) Suppose the casino limits the maximum payout to 10 billion dollars ( 10^(10) dollars). How would this cap affect your decisions in parts (b) and (c)?
Problem 1. (St.Petersburg Paradox) Consider a coin-tossing game where a player flips a fair coin until a head appears for the first time. The payout for the player is determined by the number of tosses needed to achieve the first head. Specifically, the payout is $2 if a head appears on the first toss, $4 if on the second, and in general, the player wins 2 dollars if the first head appears on the k-th toss. Each player must also pay a fixed entry fee of M dollars to participate.
(a) Calculate the expected value of the player's winnings from this game.
(b) Given an entry fee of M = 30, would you choose to participate in this game? Explain your reasoning.
(c) If the entry fee is increased to M = 1000, does your decision to participate change? Explain your reasoning
(d) Suppose the casino limits the maximum payout to 10 billion dollars (1010 dollars) How would this cap affect your decisions in parts (b) and (c)?