Probability offers us a look into the eternally stable side of life from our own jumbled world. The vehicle for this viewing is expected value. Expected value takes all the possibilities and tells you not what will happen, but rather, what happens in the long run. For example, if you flip a coin, your expected value would be 0.5 heads and 0.5 tails.

The St. Petersburg Paradox-

According to the Stanford Encyclopedia of Philosophy:

The standard version of the St. Petersburg paradox is derived from the St. Petersburg game, which is played as follows: A fair coin is flipped until it comes up heads the first time. At that point the player wins $2^n, where n is the number of times the coin was flipped. How much should one be willing to pay for playing this game? Decision theorists advise us to apply the principle of maximizing expected value. According to this principle, the value of an uncertain prospect is the sum total obtained by multiplying the value of each possible outcome with its probability and then adding up all the terms. In the St. Petersburg game the monetary values of the outcomes and their probabilities are easy to determine. If the coin lands heads on the first flip you win $2, if it lands heads on the second flip you win $4, and if this happens on the third flip you win $8, and so on. The probabilities of the outcomes are 1212, 1414, 1818,…. Therefore, the expected monetary value of the St. Petersburg game is

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In simpler terms, the chances of winning the next prize decrease at the same rate as the value of the prizes increase. Thus, each subsequent prize adds one more dollar onto the expected value. This leads to a confusing conclusion:

Lets say each try costs $10. Should you play? If you spend $10 dollars total (one attempt), you can expect to win around $3. A pretty terrible deal… right?  So why does the expected value tell me its a good deal.