(St. Petersburg Paradox) Suppose a wealthy stranger offers to play the following gamewith you:• You flip a coin until it lands Tails for the first time.• Afterward, you will receive $2 if the game lasted for 1 round, $4 if it lasted for 2 rounds,$8 if it lasted for 3 rounds, or... In general, you will receive $2 if the game lasted forn rounds.(a) What is the PMF for N = the number of rounds this game would last? What is the expected value of X = 2N , the amount of money you would win?(b) Suppose that you know ahead of time that the wealthy stranger will flee ifthe game lasts for more than 50 rounds (i.e. if you land Heads 50 times in a row, theman will interrupt the game and flee without giving you anything at all). What is theexpected value of X? Suppose that instead, the man will limit your winnings to $2.50, i.e. if youland Heads 50 times in a row, they will interrupt the game, pay you the agreed-uponamount of money, and then flee. What is the expected value of X?(c) Can you explain, in layman terms (especially without referring to convergence/divergence of series), why the expectation of Part B is significantly larger thanthose of Parts C and D?
(St. Petersburg Paradox) Suppose a wealthy stranger offers to play the following game
with you:
• You flip a coin until it lands Tails for the first time.
• Afterward, you will receive $2 if the game lasted for 1 round, $4 if it lasted for 2 rounds,
$8 if it lasted for 3 rounds, or... In general, you will receive $2 if the game lasted for
n rounds.
(a) What is the PMF for N = the number of rounds this game would last? What is the
(b) Suppose that you know ahead of time that the wealthy stranger will flee if
the game lasts for more than 50 rounds (i.e. if you land Heads 50 times in a row, the
man will interrupt the game and flee without giving you anything at all). What is the
expected value of X? Suppose that instead, the man will limit your winnings to $2.50, i.e. if you
land Heads 50 times in a row, they will interrupt the game, pay you the agreed-upon
amount of money, and then flee. What is the expected value of X?
(c) Can you explain, in layman terms (especially without referring to convergence/divergence of series), why the expectation of Part B is significantly larger than
those of Parts C and D?
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