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I was reading about St.Petersburg paradox, and understood the proof that $\frac{S_n}{n\log n} \overset{P}{\rightarrow}1$. The textbook then quotes Bernoulli:

"There ought not to exist any even halfway sensible person who would not sell the right of playing the game for 40 ducates (per play).” If the wager were 1 ducat, one would need $2^{40} ≈ 10^{12}$ plays to start to break even.

I don't understand the language (too many negatives). What was Bernoulli implying? That its ridiculous to play the game? If so, why? I didn't gain any insight from the convergence in probability result, other than the fact that you should pay $\log n$ per play if you want to play $n$ times.

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Bernoulli was claiming that nobody would pay \$40 to play a lottery whose expected value is positive, but whose chance of coming out ahead is only one in a trillion.

Of course, Bernoulli was completely wrong, as in 2014 plenty of people play the lottery every day, and the expected value of that activity is distinctly negative. The likelihood of winning is better than one in a trillion, but I'm sure that this fact does not significantly impact the players.

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  • $\begingroup$ Haha, true.. :) $\endgroup$ – elexhobby Dec 14 '14 at 0:08
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    $\begingroup$ There is a rather large difference between those lotteries and the St. Petersburg game: those lotteries do have winners. $\endgroup$ – Robert Israel Dec 14 '14 at 2:00
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If I were to play the game, the probability is very high that I would have to play a very, very large number of times before I would make money. Even though if I play long enough, I'll do well. Putting an economic framework on this, the cost of playing long enough to make a profit is too high. I probably won't live that long.

Rewording without the quote without the negatives:

"Any halfway sensible person would sell the right to play the game for 40 ducates (per play)"

parsing the quote a bit closer: Consider any even halfway sensible person who would not sell the right to play for 40 ducates. Such a person does not exist. (therefore either he would sell the right, or he is not even halfway sensible.)

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  • $\begingroup$ Thanks for simplifying the language, now it makes sense. However, this fact (you will need to play many games before you can make money) doesn't follow from the convergence in probability result right? Because the result doesn't tell you anything about the rate at which $S_n$ starts getting close to $n\log n$. $\endgroup$ – elexhobby Dec 14 '14 at 0:10

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