Playing va banque

Mr Smith and Mr Brown play "heads and tails" using a non symmetrical coin. Heads drops out statistically 52 in 100 times. Mr Smith has got $100 and bets heads, staking $1 on each bet. After 1000 rounds statistically Mr Smith will win 520 times on average and lose 480 times. That makes a $40 profit. What if he wants to win more?

He can take a risk and bet all the money he has. After five games won in a row he would have $3.200. It would be a profit of $3.100 net. It is quite a profit but the risk of going bankrupt is enormous- the chances of winning five games in a row are less than 4%.

Staking one`s all on a chance is something that desperate players would do.

On the other hand , repeating the same game a thousand times, hoping for $40 profit, would irritate even the most patient player.

Is there a golden mean then?

Mr Smith could bet more than $1 and less than $100, each time leaving some money aside just in case – to avoid going bankrupt and having a chance of winning back in the next round.

If a fraction of risked money is high (i.e.60%), then the results of the initial games would decide about the financial balance.

If the loses predominated in the beginning, he would lose a considerable part of his capital. He would not be able to win it back in the next part of the game. Thus, he would finish the game with less money than what he had at the start.

On the other hand, using a very low percentage of the player`s capital in the further games (i.e. 2%) would be a safe strategy. However, this kind of security would not let the player win a considerable sum of money.

Now the question is: how to bet (what percentage of the capital should be invested in the further games) to maximalize the profit?