Question

the chance of winning a bet is 50%. If you win a bet, you receive the same amount that you put in. you plan on betting in a "double-down" loss scheme of $5/$10/$20/$40/$80/$160 for each consequtive loss. This equates to a total of $315. If you were to win a bet, the following bet will be $5. for example, if you lose 3 times on a row and win one the fourth, you would bet $5/$10/$20 /$40 and then the next bet return to $5. From this method, so long as you win at least one time prior to losing 6 in a row, you will win $5. However, if you were to ever have a 6 loss streak, this would mean you lose the $315 you started with. in order to not be at a defficiet, you would need to win at least 64 times prior to losing 6 in a row to reach $320 profit. this would mean that further wins are profit to the initial investment, even if one were to eventually receive a 6 loss streak. what is the probability of you profiting? in other words, what is the probability of winning 64 times without losing 6 times in a row

Answer 1

The player should have to win at least one game in the six consicutive games. If he lose the 6 games in a row then the the player will lose i.e $315 that is he started with and he can't play any further because he don't have any money left to play after 6 loss streak.so Given that "winning chance is 50 percentage" they means loosing percentage is also 50 percentage only (100-50=50). Which means 0.5 probability . It means probability of losing 1 game is 0.5

Therefore probability of 6 lose streak is

if the player wins at least one game in the 6 games that are in a row, he will not be eliminated and there will be a chance for 64 wins.so there should be at least one win in every 6 consicutive games. So he can continue the game

Therefore probability of winning at least one game in a 6 consicutive games is

like this the player should have to win 64 times without 6 games lose streak that means at least one win in 6 consicutive games like this 64 times

Therefore probability of winning 64 times is 0.36. we can observe in the following pic.

the key point here is the player should have to win 64 times like at least once in every 6 consicutive games upto 64 times :)