Sample Question #41 (probability theory)

You are offered to play a game of chance. A fair coin is tossed repeatedly until you get the first tails, at which point the game ends and you get the prize. The prize "pot" starts at $1 and doubles each time you get heads. So for instance, if you get heads the first toss, the pot becomes $2. If you get heads again the second toss, the pot becomes $4. If you get heads the third time, the pot becomes $8. If the fourth toss gives you the tail of the coin, you win and take home the $8 prize.

Before you play, you must pay a fee to enter this game. The question is, what’s the maximum amount you’re willing to pay in order to play this game? Explain your answer carefully.

(Hint: if you studied probability theory, you would have recognized this as the famous St. Petersburg Paradox – it’s just a great puzzle, isn’t it?)

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ANSWER

Expected payoff = 2*(1/2) + 4*(1/4) + 8*(1/8) + 16*(1/16) + … = infinity. So the maximum amount you should be willing to pay is infinity! Hence this is a paradox.

But in reality, you wouldn’t pay that much. The reason is you have a risk-averse utility function, so you’re afraid of the big loss from paying too much to play the game.