A gambler offers you the following deal. You have to keep tossing a fair coin until you get a heads, at which point you stop and collect your winnings: if it happens after n throws, the gambler will give you $2^n$ cents.
However, in order to play the game you first have to agree to pay an entrance fee, but the amount you need to pay is not known to either of you. Instead, you each ask a stranger (before the game starts) to secretly write a random amount of money on a piece of paper and seal it in an envelope.
Once the game has ended, and having paid you your winnings, the gambler will open both envelopes and you then have to pay him the average of the two amounts that the strangers have written (rounded up to the nearest penny).
From a financial point of view, is it worth playing the game?
$$ P(getting \space heads)=P(getting\space tails)=1/2 $$
If we obtained head after say, 5 throw, then it will be $2^5$ cents
This question is seriously baffling me. There's little to no numerical information that I can go on besides the 1st paragraph.
Though from overall looks of it, it seems that this is a Markov chain process on a countable state space that uses a random walk.
I'm completely lost. Could you help me out.
With respect to the game, this is exactly the St. Petersburg paradox.
reference:
http://en.wikipedia.org/wiki/St._Petersburg_paradox
The game has infinite expected value but is worth only a finite amount in practice if that makes sense?