So there is a person A and a casino C, each rolling a cubical fair die. The one who obtains the highest number wins. If they both obtain the same number, then the casino wins. The player bets $a$ and the casino offers a payout multiplier of $m$. (And it is easy to show that $m<2.4$ if the casino is to make profit.)

I am okey with all this, but if the player plays more than once, does the casino make different profit depending on the number of times the player plays, $k$. Clearly, I would say that as the game is repeated the casino will win more on average (since it has greater odds of rolling a highest number) but a friend of mine suggested that "if the player stays longer than a given number of games, no profit is being made."

I don't understand this. Each time the player plays it is like being a different player, so nothing is really complicated. So how does it make a difference, for the player's and the bank's profit, if the player stays for $k$ games.

I know that if the player plays the game $k$ times the probability of winning $x$ times is binomially distributed $$P(X=x)=\binom{k}{x}\left(\frac{15}{36}\right)^x\left(\frac{21}{36}\right)^{k-x}.$$

Thanks in advance.


Your thinking that each bet is independent and hence playing multiple times does not make the expected value of each turn different is sound.

However there is a possibility that your friend referred to a betting system - e.g. you bet larger each time, say a multiple $b$ of previous bets, and then if you have unlimited funds, unlimited time, no limits on bets etc., you could eventually expect to win, when you can more than make up for all losses so far by proper choice of $b$.

Search for St. Petersburg Paradox or Martingale betting if you need more on this.


If you play a martingale strategy, you need an infinite bankroll (although a bankroll much bigger than the casino's will usually work)


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