Sorry! Very sorry!
I miscalculate so forget this thread.
Here is the challenge.
We flip a coin 100 times.
If the outcome is equal to the bet (k dollars) the gambler receive 2*k dollars (return = 1 k). If not the gambler loose k.
I simulated the game with 2 gamblers : John and Paul.
John bets always head.
Paul always tail.
But each one of the players bet different sum of money.
John bets a(i) dollars and Paul b(i) dollars.
i is the index of each toss.
i vary from 1 to 100.
John and Paul start with a bankroll of 200 dollars each.
Can you find the 2 bet sequences a(i) and b(i) such as the 2 players have 95% chance to make BOTH a profit of at least 1 dollar? They stop betting once their profit is equal to at least 1 dollar
I have found a solution. What about you?
The simulation was right. Here is my problem : there is ALWAYS during the 100 tosses an interval where the value is positive (more than one dollar) if the 2 players bet the right sequences a(i) and b(i). Now is there a way to compute when to leave the game? When some positive value is reached? Which value? The 2 players have 38% chance to loose during 10 consecutive tosses. If we can find when to stop the bet then any player can bet for 2 fictive players and win every day 1000 dollars or more (depending on the bet limit fixed by the casinos). I made an exhaustive study of 10 consecutive tosses and I expect that with 100 tosses the loss probability will decrease. I want you to help me to make an exhaustive study of at least 30 consecutive tosses. The bet sequence a(i) is 1-2-1-2-1-2.... and the bet sequence is 2-1-2-1-2-1.....It is my result but you can do better. Thank you for any clue.