### Brain Teasers

# St. Petersburg Paradox

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

You are offered a game to play with a single fair coin. It costs 20 dollars to play this game, but you can win much more than that. The way it works is that you continue to flip the coin until you get tails. For every heads you get before that, your payoff doubles. For example, if you get:

Heads

Heads

Tails, then you would earn 4 dollars.

In other words, you get: 2^heads dollars after you play. The question is: would you come out with more or less money after you played this game an INFINITE number of times? Remember, each game costs 20 dollars!

Heads

Heads

Tails, then you would earn 4 dollars.

In other words, you get: 2^heads dollars after you play. The question is: would you come out with more or less money after you played this game an INFINITE number of times? Remember, each game costs 20 dollars!

### Answer

Neither!You would come out with an INFINITE amount of money! Here's why:

The way to calculate an expected value of a game=(the probability of event1)*(the payoff from event1)+(the probability of event2)*(the payoff from event2)...

Let's say:

event1=Tails

event2=Heads,Tails

event3=Heads,Heads,Tails, and so on.

The probability of these events are:

event1=1/2

event2=1/2*1/2=1/4

event3=1/2*1/2*1/2=1/8, and so on.

The payoff of these events are:

event1=1

event2=2

event3=4

event4=8, and so on.

Plugging this into the expected value formula, we get:

EV=(1/2*1)+(1/4*2)+(1/8*4)+(1/16*8)...

This simplifies to:

EV=1/2+1/2+1/2+1/2...

Any number added an infinite number of times will sum to infinity, so your expected value of this game is infinity.

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