A person tosses a fair coin until a tail appears for the first time. If the tail appears on the $n$th flip, the person wins $2^n$ dollars. Would you be willing to pay $1$ million for each game if you could play as long as you liked and only had to settle up when you stopped playing?
I thought about this question and ran a computer simulation, except with the entry fee being $100$ dollars each round and ran $100000000000$ rounds, and stopped if there was any profit. In almost each run, there seemed to be a net profit. However, I couldn't mathematically produce the same result.
I read the following quote: "But after a very long time, you will win so much money that is sufficient for you to pay off this large debt and still purchase the whole world. As @IanColey put it, this is because the chances of winning so much money are very, very tiny, but the payoffs associated with these very, very tiny probabilities are much, much, much more enormous than the probabilities are tiny.
I couldn't fathom that one single round would be able to earn so much money to pay off the debt that had been accumulating. So I tried making an equation for Money Earned - Money Spent. Money Spent = $1000000k$, where $k$ is the number of rounds played.
Expected Value[Number of Flips to get $a$ heads in a row] = $E_a =\frac{E_a+1}2+\frac{E_a+2}4+...+\frac{E_a+a}a+\frac a{2^a} = 2(2^a-1) $
Since tails is expected to occur on the second flip, the expected number of flips in $k$ rounds is $2k$. Now, we have $a = \log_2(E_a+2)-1$ and you win $2^{n+1}$ dollars in a round, where $n$ is the number of heads before the first tails. Therefore, you have the expected highest winning after $k$ rounds to be $2^{(\log_2(2k+2)-1)+1} = 2k+2$.
Therefore, after $k$ rounds, you would be expected to lose $9999998k -2$ dollars.
What am I doing wrong? I know a huge oversight in this math is to only account for the total gain for one (largest) flip. However, I made this assumption based on the above quote that after a while, you will win so much that you will pay off the large debt.
Half the time, tails happens first, you receive $\$2$, which averages out to $\$1$ per game.
A quarter of the time, tails happens second, you receive $\$4$, which averages out to another $\$1$ per game.
You get another dollar per game from the times tails is third, and fourth, and so on.
Your $100000000000$ games is around $2^{37}$, so the first $\$37$ per game is likely to appear. Any more than $37$ is a bit hit and miss. Try charging $\$30$ per game, you are likely to see a profit over $100000000000$ games.