- #1
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- Homework Statement
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- Relevant Equations
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I feel like there is something missing in my solutions, because the answers are coming out weird.
(a)
The reward for getting a head is ##2^n## dollars, the number of flips ##n## it takes to get the first heads.
the probability to get a head in the ##n##th flip is ##\frac{1}{2^n}##
so the expected value of the game is
$$E(X)=\frac{1}{2}\cdot 2+\frac{1}{4}\cdot 2^3+\frac{1}{8}\cdot 2^3+\frac{1}{16}\cdot 2^4+\cdots$$
$$E(X)=\sum_{i=1}^\infty 1 \Rightarrow \text{ the sum diverges}$$
Who would want to pay an indefinite amount of money to play this game...?
(b)
$$E(X)=\sum_{i=1}^{40} 1 = 40$$
By this logic, I would pay no more than 40 dollars to play this game
The probability of going more than 40 rounds is
$$P(X>40)=\sum_{i=41}^\infty\frac{1}{2^i}\approx 0$$
Yet, the expectation value drastically differs from the value in (a)
(c)
$$E(X)=\sum_{i=1}^{40} 1 + \sum_{i=1}^\infty \frac{1}{2^n} =40+1 =41$$
I'd pay 41 dollars to play this game
(d)
no one has that much monay. that's too many zeros.
It turns out this is famous problem called St. Petersburg Paradox. - it seems to be a rare example of when probability theory isn't rational for predicting the outcome, since the player is probably only going to win a modest amount of money in this game.
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