Here is a write up I have done ages ago for martingale:
Lets say you want to play martingale and cover at least 7 losses in a row starting from $5 bet. So the money you need to make the 8 bets is $5+$10+$20+$40+$80+$160+$320+$640 = $1,275.
Let us use the martingale progression on roulette betting on red. The chance of winning is 18/37 (48.649%) and loosing is 19/37 (51.351%). So the chance of loosing my 8 bets is 0.51351^8 = 0.004835 (0.4835%). In other words I have a 100%-0.4835% = 99.517% chance of winning $5. So its extremely likely that I will win $5 each time I attempt the martingale, and extremely unlikely for me to loose $1,275. Infact the loss of $1,275 comes once every 1/0.004835 = 206.83 times I try to win $5. So the risk is always far more than any possible reward. In other words on average I will win 206*5=$1030 for every $1275 I Loose.
Betting on red is an even money bet. So I want to double my $1,275. That’s 1275/5 = 255 lots of $5, but as I said above the loosing streak comes up once every 206 bets. The chance of winning $1275 starting from $1275 is 0.99517^255 = 29.09%. This is why the martingale does not work.
So starting with $1275 you only have a 29.09% chance of double your money. While simply betting it all on red in one spin is 18/37 = 48.649% chance. So using martingale will actually loose money faster. This is because the average bet size on martingale progression is much larger than just flat betting $5. If you are going to play martingale you would be by far better off just betting all your money on one spin and calling it a day, win or loose.
Most people claim that martingale can win, if you avoid the bad streaks. But of course you never know when that bad streak will happen, so you cannot avoid it. Remember that extending any streak by 1 is the same odds as 1 streak. For example if you wait until black has spun up 5 times, then start the martingale, you will still fail. This is because a streak of 15 losses is just as likely as a streak of 10 after you have already waited for a streak of 5.