### 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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## Comments

weird

Yes, but you can never reach infinity, so I would have said "a large number of times", for clarification.

Oh well. Good teaser!

Oh well. Good teaser!

So what is the answer? the question is "would you come out with more or less money" and the answer "infinate" does not answer the question asked. is Infinate more or less?

The trick is, an infinite number of times. This assumes you have sufficient money to buy the chance to play and then continue on playing.

You know those large casinos in Las Vegas that cost billions? They all get paid off. You know why, because no one can play an infinte number of times and the house does not pay true odds!

You know those large casinos in Las Vegas that cost billions? They all get paid off. You know why, because no one can play an infinte number of times and the house does not pay true odds!

I disagree with the payout, that is. While it is true that the payout is $1, $2, $4 etc, it cost $20 to play the game. While your total payout is infinite, so is the COST to play the game. The net payout is only $1-$20, $2-$20, $4-$20 etc. Putting this into the equation for payout you end up with an EV of -$9.50 for the first factor and slowly increases, but does not become positive until the 6th factor. If you sum the event values for events 1-n, the net value does not become positive until n=41. In other words, you'd have to be able to flip 40 heads in a row before flipping a tail. Odds of that are less than 0.000000000001. If you can afford to play that many times, then yes I agree it would start to pay off. But mind you, that would be a payout of $550 billion and that is just to break even. So unless you've got that kind of cash laying around, DON'T PLAY!

yes, you're absolutely right... but i said an INFINITE amount, and that assumes you have enough money.

the expected value is for an infinite game, any practical attempt to play this game would result in you losing, or winning a not so worth it amount of money so personally id never play it for any ammount. much rather go counting at the blackjacks :p

SO the odds of the dun rising blue and you gfet paid double on the moon = infinite?

THe whole wording of the question was vague and confusing, the answer possibly even more so.

The pay out doubles buit sinbce the payout was never stated that doesnt help alot. True odds would be 20 for 20. T he given example would then be -20 + 40 - 20 + 40 - 20 = 20

THe answer of infinte is obviously wrong in an event. THe question was more or less, therefore the answer to a real question needs to be one or the other and if you say after the fact that you start with infinite money then of course I would always end up with less.

I start with the most posiible so can never get more, so the only option would be less.

THe whole wording of the question was vague and confusing, the answer possibly even more so.

The pay out doubles buit sinbce the payout was never stated that doesnt help alot. True odds would be 20 for 20. T he given example would then be -20 + 40 - 20 + 40 - 20 = 20

THe answer of infinte is obviously wrong in an event. THe question was more or less, therefore the answer to a real question needs to be one or the other and if you say after the fact that you start with infinite money then of course I would always end up with less.

I start with the most posiible so can never get more, so the only option would be less.

Sorry I misunderstood the beginning of the game,"you flip until you show tails" I thought each flip cost 20. But still with no stated payout it woudl be impossible to figure.

Its good to know however that at least we chose to play a game where we coudl win more than the cost of the game.

But the fact remains the answer is impossible since if I paly the game an infinite number of times I will never be done playing so I willnever have a final result.

ANd if I start wih infinite money what is the motivation to gamble anyway or look to increase my money?

Its good to know however that at least we chose to play a game where we coudl win more than the cost of the game.

But the fact remains the answer is impossible since if I paly the game an infinite number of times I will never be done playing so I willnever have a final result.

ANd if I start wih infinite money what is the motivation to gamble anyway or look to increase my money?

What the ????????????? SourDough only likes Disambiguous Questions and Teasers, or DQT and you kids say!

$D

$D

I knew what answer you were probably looking for, but as soon as I read "after" and "infinite" I knew the answer was "neither" since "after infinite" will never occur.

I can't believe I caught you saying something incorrect about math, javaguru! Your statement is only true if you stipulate that playing one game takes a non-zero amount of time.

After all, just ask Zeno -- you complete an infinite number of tasks every time you move. The trick is that the tasks must take (on average) zero time. (I know javaguru won't need to, but anyone else who wants to understand this, look up Zeno's paradox on wikipedia.)

But, since you could never make this wager in zero time, I agree you could never do it an infinite number of times.

After all, just ask Zeno -- you complete an infinite number of tasks every time you move. The trick is that the tasks must take (on average) zero time. (I know javaguru won't need to, but anyone else who wants to understand this, look up Zeno's paradox on wikipedia.)

But, since you could never make this wager in zero time, I agree you could never do it an infinite number of times.

OK, you got me. Poorly worded statement of mine as it relates to pure math.

In the real world, however, the evidence suggests that space and time are discrete and not infinitely divisible, that there is in fact a quantum period of time and a quantum unit of distance. This means that all longer periods of time or greater distances are multiples of these quantum units. These size limits are known as the Planck constants and are derived from properties of light (photons). The unit of distance is 1.6 x 10^-35 meters and 5.4 x 10^-44 seconds.

So Zeno's paradox doesn't exist in the real world, and can only exist in a mathematical approximation of the real world.

In fact, the strongest argument in favor of time and space being discrete is the havoc that infinity causes with Einstein's equations of relativity. Black holes can't really have infinite density as the theory of relativity predicts. The Planck density is on the order of a googol (10^100) grams per cubic meter, which is pretty darn dense, but falls far short of infinite.

In the real world, however, the evidence suggests that space and time are discrete and not infinitely divisible, that there is in fact a quantum period of time and a quantum unit of distance. This means that all longer periods of time or greater distances are multiples of these quantum units. These size limits are known as the Planck constants and are derived from properties of light (photons). The unit of distance is 1.6 x 10^-35 meters and 5.4 x 10^-44 seconds.

So Zeno's paradox doesn't exist in the real world, and can only exist in a mathematical approximation of the real world.

In fact, the strongest argument in favor of time and space being discrete is the havoc that infinity causes with Einstein's equations of relativity. Black holes can't really have infinite density as the theory of relativity predicts. The Planck density is on the order of a googol (10^100) grams per cubic meter, which is pretty darn dense, but falls far short of infinite.

As stated earlier, the cost is extremely important too. You do not explain how the cost factors in, in your explanation, so it is not actually valid, though your answer is correct. Just that the payout is infinite does not mean that the profit is infinite. Consider this conversation:

Bob: I sell these radios for $8 each. Imagine if I could sell infinite radios! That would be great! I'd be infinitely rich!

Paul: How much are you buying them for?

Bob: $10 each. But still, it's infinity times $8 of income. Infinite income = infinitely rich.

Paul: ......

Bob: I sell these radios for $8 each. Imagine if I could sell infinite radios! That would be great! I'd be infinitely rich!

Paul: How much are you buying them for?

Bob: $10 each. But still, it's infinity times $8 of income. Infinite income = infinitely rich.

Paul: ......

No, the cost is actually not relevant. The expected payout of a single game is infinite, which suggests that it's worth any (finite) amount of money to play the game, especially if you're going to play infinitely many times.

The cost becomes relevant when you're looking at playing a finite number of games. With the cost at $20 you have a 31/32 chance of losing money on any given game, even though the payout on that 1/32 chance is potentially huge. The more you play, the more chance you have of scoring that big payout to cover the losses.

The cost becomes relevant when you're looking at playing a finite number of games. With the cost at $20 you have a 31/32 chance of losing money on any given game, even though the payout on that 1/32 chance is potentially huge. The more you play, the more chance you have of scoring that big payout to cover the losses.

The paradox here is that the game appears to have an infinite expectation value, whereas in practice nobody would actually pay more than a modest amount to play. The problem is that the conditions of the game are unrealistic - you can't play an infinite amount of times nor is there an infinite amount of money in the world to be won. One simple resolution is to fix a limit for the amount of money that the bank has. This causes the expectation value for playing a game to converge to a finite amount.

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