We have a coin that has a probability $p>1/2$ of coming up heads (and probability $1-p$ of coming up tails). We now play the following game:
- We start with a fortune of one dollar.
- We toss the coin. If it comes up heads, we double our current fortune, and we repeat this step (step 2). However, if it comes up tails, we lose all of our fortune, i.e. go broke, and stop playing.
Note that after $n$ plays, our fortune is $2^n$ with probability $p^n$, and zero with probability $1-p^n$.
This means that the expected value of our fortune after $n$ plays is $2^np^n = (2p)^n$. Since $p>1/2$, we have $2p>1$, which tells us that our expected value after $n$ plays approaches infinity as $n$ approaches infinity.
On the other hand, our probability of going broke after $n$ plays, $1-p^n$, approaches one as $n$ approaches infinity. In other words, we know for sure that we will eventually go broke.
My question is this: How can we, in the infinity, both possess an infinite fortune and yet be completely broke? How can this paradox be resolved?