I have thought up a paradox, that may already exist, but I do not know what it's called. It's bothering me though, so any help regarding solving it or proving it impossible would be appreciated.
In this paradox, you have a gambler. The gambler is \$200 in debt, but the gambler has a wealthy friend who lets the gambler bet using money he does not have to play a game. In this game, the gambler bets \$1, and a random number generator generates a number from 1 to 100. If the number roll over, or exactly 55, the gambler wins \$2. If the number rolls under 55 the gambler loses that \$1 he bet.
From my understanding of statistics, over a certain expected period of time, the gambler should hit the unlikely scenario to get out of debt using this method. However using my computer simulations, it seems to take more time than I can allow my simulations to run.
Is it possible to guess an expected number of times the gambler would have to play the game in order to get out of debt using some
mathematical model?
I am also concerned that the nature of random-number generators may make it impossible for the gambler to get out of debt, as random number generators could be heavily biased to avoid situations such as randomized decks to be fully sorted, or getting out of debt with negative debt and unlikely odds.
What I want to get out of this question is, how to explain why it's possible, how to calculate expected number of times the game has to be played to reach the goal, or why it's impossible, or some existing question I can try to study to better understand the problem.
Best Answer
This has little to do with random number generators.
As Tony K noted, this is a random walk where each step is $+1$ with probability $p < 1/2$ and $-1$ with probability $1-p$. Starting at $0$, the probability of ever reaching positive integer $m$ is $(p/(1-p))^m$. In your case, with $p = 0.46$ and $m = 200$, that probability is about $1.18 \times 10^{-14}$.