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St Petersburg Paradox

The Bernoulli Paradox is a probability puzzle that dates back to 1713 when Nicolas Bernoulli first conceptualized it.

It eventually became known as the St Petersburg Paradox when his cousin Daniel Bernoulli published his thoughts on it in: Commentaries of the Imperial Academy of Science of Saint Petersburg.

This paradox is centered around a seemingly simple coin-tossing game, which appears to have an unusual theoretical outcome...

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St Petersburg Paradox

The game consists of tossing a fair coin repeatedly until it lands on tails. The player's winnings are determined by the number of coin tosses it takes for tails to appear, with the payoff being €2ⁿ, where n is the number of tosses.

Thus, the player wins:

2 euros if tails appears on the first toss,
4 euros if heads appears on the first toss and tails on the second,
8 euros if heads appears on the first two tosses and tails on the third, and so on.

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Running the Trials

Round	Coin Tosses (n)	Earnings (€)
1
2
3
...
10

Calculate your average earnings...

What would happen to your average earnings be if we kept playing?

Play the St. Petersburg game with your partner for 10 rounds each. Record the results in a table recording your winnings.

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Theoretical Values

Number of trials before Tails	Story...	Probability	Winnings
1	T	½	€2
2	H and T	½ x ½	€2²= €4
3	H and H and T	½ x ½ x ½	€2³= €8
4
5
6
7
8
9
10

Copy the table into your copies and complete the missing celss

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Expected Value

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Expected Value

€2 (1/2) + €4 (1/4) + €8 (1/8) + €16(1/16) +...+ €2ⁿ (1/2)ⁿ

€1 + €1 + €1 + €1 + ...+ €1

As the value of n increases the expected value tends towards infinity…

How does this compare to your actual average value?

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How much would you be willing to pay to play the St. Petersburg game once?

Explain your reasoning.�

Let’s play

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ChatGPT – What would you be willing to pay?

Personally, I would be willing to pay a moderate amount, say €5 to €10, to play the St. Petersburg game once. My reasoning is based on the following factors:

Risk aversion: Although the theoretical expected value of the game is infinite or extremely large, the probability of winning a very high prize is quite low. Most of the time, the game will end after just a few coin tosses, resulting in relatively small winnings. As a risk-averse individual, I would prefer not to gamble a large sum of money on a low probability event.
Diminishing marginal utility: The utility or satisfaction I would derive from winning a large sum of money decreases as the prize gets larger. For instance, the difference in utility between winning €1,000 and €2,000 might be significant, but the difference between winning €1,000,000 and €1,001,000 would be much less noticeable. This diminishing marginal utility influences my willingness to pay for a game with such extreme possible outcomes.
Practical considerations: In real-life situations, there may be constraints on the amount of money I have available to spend on a gamble or my willingness to accept risk. Additionally, the game's setup might have limitations, such as a maximum prize payout, which would further reduce the expected value.

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