I was just thinking about something to do with gambling, and it occurred to me that surely you could guarantee success at a gambling game in the long run if you set aside profits.
Allow me to explain. Say you had 4 betting units.
If you were to flat bet one 1 unit each round, but set aside any profit that you made on top of your original 4 units. When you reach 4 units in profit or loss, (disregarding any previous winnings), you quit.
Therefore, if we assume there is a 50% chance of either winning or losing your 4 units, surely there is also a chance you will near your 4 unit profit goal, but then lose your original 4, leaving you with 3 units. Therefore, your loss for that session is -1.
Therefore, there are 4 different ways one can lose. Having lost 4 units but gained 3, lost 4 gained 2, lost 4 gained 1, or lost all 4 straight away.
If it is 50/50 that you will win or lose 4 units, then surely by safeguarding profits, then the possible outcomes would be, +4, -4, -3, -2, -1. Because of this surely the 50% that you lose must be divided up again to consider losing only 3 or 2 or 1 unit.
Surely then that means the probability of each outcome is: +4 : 1/2 -1 : 1/8 (1/2 divided by all negative outcomes.) -2 : 1/8 -3 : 1/8 -4 : 1/8
All the probabilities add to 1 as should be expected. But when calculating expected value we get:
+4 * 1/2 + -1 * 1/8 + -2 * 1/8 + -3 * 1/8 + -4 * 1/8 = 3/4
This shows a positive expectation, which should not be possible.
Can anyone tell me the flaw in my reasoning or workings, as this is puzzling me greatly.
Your chance of ending at $+4$ is not $50\%$. The symmetry is broken when you refuse to bet your winnings. If you use the stopping criterion of stopping when your net is $+4$ or $-4$ your chance of ending at $+4$ is indeed $50\%$, but in your system a series of $WWLLLL$ would cause you to stop and you could never reach $+4$ that way. That reduces the chance you will reach $+4$.