i am stuck on this problem Problem 6 (St Petersburg Paradox). Suppose you have the opportunity to play the following game. You flip a fair coin, and if it comes up heads on the first flip, then you win $1. If not, then you flip again. If it comes up heads on the second flip, then you win $2, and if not you flip again. On the third flip, a heads pays $4, on the fourth $8, and so on. That is, each time you get tails, you flip again and your prize doubles, and you get paid the first time you flip heads.a) How much should you be willing to pay to play this amazing game? In other words, compute the expected payout from playing this game.For Part A, would the E(winning)= infinity?b) Now suppose the casino (or wherever you’re playing this game) has a limited bankroll of $2n. So, if you get tails n times in a row, then the game is over automatically and you are paid $2n. Now what is the expected payout? How much should you be willing to pay to play the game if n = 10? For Part B, would it be the same answer as Part A?

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Asked Sep 19, 2019
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i am stuck on this problem

 

Problem 6 (St Petersburg Paradox). Suppose you have the opportunity to play the following game. You flip a fair coin, and if it comes up heads on the first flip, then you win $1. If not, then you flip again. If it comes up heads on the second flip, then you win $2, and if not you flip again. On the third flip, a heads pays $4, on the fourth $8, and so on. That is, each time you get tails, you flip again and your prize doubles, and you get paid the first time you flip heads.
a) How much should you be willing to pay to play this amazing game? In other words, compute the expected payout from playing this game.

For Part A, would the E(winning)= infinity?

b) Now suppose the casino (or wherever you’re playing this game) has a limited bankroll of $2n. So, if you get tails n times in a row, then the game is over automatically and you are paid $2n. Now what is the expected payout? How much should you be willing to pay to play the game if n = 10?

 

For Part B, would it be the same answer as Part A?

 

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Expert Answer

Step 1

As per the question,  Table is drawn. On First head you get 1 dollar, on second head you get the 2 dollar and on 3 you get 4 and so on.

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Trials when head Probability P(x) Рayout(x) оccurs 1 1 1 2 1 2 2 22 1 3 23 1 24 So on So on So on t

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Step 2

Expected payout is given by

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Е(X) — Хх х р(х) — 1 х-+2 х 4 x 8 x So on 1 1 1 1 So On 2 2 2 2 EOO + +

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Step 3

b)

Now here we have given n=10

Let n denotes the no of t...

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P(n) Prize Expected Pay off 1 21 2 1 1 2 1 22 =4 1 2 22 1 23 8 3 1 23 1 24 16 4 1 24 25 32 1 25 26 64 6 1 26 1 27 128 7 1 27 1 28 256 8 1 28 1 29 512 1 29 1 210 1 10 210 1024 C O

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