The Truth about Betting Systems - Wizard of Odds

ELI5: The math of chance / streaks. (Bernoulli Process?)

So everyone who has taken a statistics class has heard the old "You've flipped a fair coin 4 times, and gotten heads every time. What are the odds that the next flip will also be heads"?" question. And I know that the answer is 50% - because the coin retains no memory of what happened before, so it doesn't matter what the previous 4 flips were. Or previous 1 or 100.
That math makes sense to me. However. There's a nagging underlying unanswered question that one is hunting at, but not answering. The "common sense/gut" experience that makes you not answer that correctly is wrong because somewhere common sense says that the true odds of repeatedly hitting the exact same result over and over when given the same 50% chance with each role. Obviously in reality that has got to happen somewhat less often than 50%. I'm confident enough in this theory that I'd even go ask far as proposing that the chances of longer streaks are even lower than shorter ones. But my logical brain still wants to talk numbers.
I know that mathematically, once randomness is brought in, that things get a bit fuzzy. I know there's no true hard number for calculating pure chance, (or luck, which is effectively what a streak is.). And at the end it doesn't matter because over an infinite number of attempts, the result will ALWAYS come to rest at 50%.
But. What if there isn't an infinite number of attempts? Because that's the world as we experience it.... and there are limitations hard coded for us (i.e. time / a limited feasible life span). With that in mind, I guess I'm asking if there is a way to calculate the odds of hitting a 50% chance 10 times in a row in, say, 500 attempts? My gut tells me the odds are lower than in 1 billion attempts, yet higher than in 20 attempts.
Does the introduction of a hard cap on attempts before stopping chance any of the calculations? And is there any way to estimate the probability with any accuracy at all before a streak starts to begin moving towards the statically inevitable "correct" probability of 50%?
I can't seem to find a good answer. But apparently this kinda stuff is what computers are good at. Based on some scripts I've found online, it seems pretty constantly that the odds of hitting heads 15 times in a row seems to be around 1:100 billion. Again. I know that result is just fuzzy data based on chance results of how lucky a computer program is. But. That 1:100 billion sure feels a bit closer to my gut than 1:2.
I first became interested in this paradox after being explained the "Martingale" gambling strategy. In short, this strategy is taking advantage of the common sense belief that streaks just happen less... So. With that theory, if you were to play a game of chance in which the odds of winning or losing are always 50/50, after each loss, you simply double your last bet, and it only takes 1 win to recoup losses, even after a long streak of losses.
In Vegas, a popular game close to 50% is Blackjack. When played "by the book", the odds favor the house by 51% vs the player's 49% odds. So, 1% of all money gambled should be profit. Clearly this over billions of bets means Casinos make a lot off Blackjack over time safely. But overall, it's profitable for the Casino by 1% or the amount bet over time.
However all these debunking "tests" are generally run in definately. So basically they will run however many times they need to to fail, as the only "stop" to the simulation is at the improbable but inevitable streak of losses. But, there's a limit to how many attempts you can try in your lifetime. And these simulations always roll earned profits unto your "available money before you go broke. So any winnings gotten before you hit that rare streak and are broke are forgotten. But what if your profits never got gambled and you always stopped after a set amount of times? I know all the stated reasons this isn't a truly failsafe strategy. And I do know that this strategy obviously will eventually hits a wall... But what are the odds it hits that wall before your winnings are significantly worth it?
So. In normal debunking explanation this plan obviously bumps into the wonderful world of compounding numbers. And yes, most people both underestimate how much money they need to have to cover realistically common streaks of 4-5 losses by doubling down at every loss. I'll never forget when I learned of compounding interest... how that number jumps incredibly at some point. It definately happens rapidly. Also, common sense also seems to forget the fact that in order to keep this going, you don't just need the cash to float your current bet - that until you hit a win, you are out the combined total of all the previous lost 9 bets as well. ($5 + $10 + $20 + $40 + $80....). So. Let's say, with a $5 bet, I believe it takes 9-10 loss streak before you'd be at a $5,160 bet. So you'd need $10,260 available to use. And it gets way worse after that.
So. As anyone who has tried this trick will tell you, it becomes unexpectedly unrealistic and the crash happens unnerving rapidly for most. Also, I imagine that anyone bothering with this is unlikely super rich already.
So, in reality, the casinos are safe. They have a mathematical shield because people rarely would have enough cash to cover the inevitable losing streak that wipes them out before whatever long term mathematical advantage can balance their risk.
But. If the magic of compounding numbers wasn't enough to protect them against this strategy, the casino piles on more little tricks to make this strategy super unrealistic. Such as, they apply a "floor" and "ceiling" on bets by making table minimum bets $5, and applying table max bets at $5,000 or so.
This means that with doubling down on at $5 bets at the least. With that, the more mathematically common (and inevitable) streaks of 5-6 losses require a person to have enough cash on hand that probably aren't into this strategy of grinding out $5 profits like a MMO gold farmer.
But. Just in case, in the rare event if a streak of 9 in a row ever hits and the casino risks a large loss, the table max is in place to effectively allow the casino "banks" it's profits at a particular amount. In short, they place a limit on anyone placing a large enough to keep doing this endlessly. There are also more "human psychology" factors added to the mix... like training dealers to mock someone using such a "flawed strategy" as soon as they notice it... while also giving an endless about of free free vodka to those playing long stretches.)
So, over time, if the math doesn't end it, humanity eventually rears it's head, and maybe a mistake is made due to exhaustion, or you accepted one too many free drinks, or you are shamed away from the table by the dealer... Something is likely to occur before doing much damage.
But in reality, after doing this for a time, you are far more likely to have not hit the killing streak yet, and up until then, you even have some profits. Maybe you hit a winning streak. Regardless, at some point doing this, you will inevitably feel what I can only describe as an OVERWHELMING greedy realization that if you had only bet $10 per hand, you'd have twice as much money. Fighting this temptation to increase your bets after winning is very hard. Just one more thing makes the Martingale strategy a practical failure. But if you account for these things - when is it a mathmatical failure?
So... between hard caps on losses, human nature (which is being encouraged by the casino). Most call the Martingale strategy "debunked". Not just by the 50% math over the long term, but by factors put in place short term. (Minimum/Maximum bet amounts, free alcohol, and even maybe some old fashioned high-school bullying by the dealer if you do to try this. Or greed.
But more real limit is amount of bets you can manage in a set time. A hand of Blackjack at a full table can take 10 minutes. So, in an hour you get 6 hands in. And took 6x your bet in profit. Not bad, but not just a boring way to "gamble", it's hardly enough profit to deal with the hassle of doing it. And the Martingale is left to rot in page 82 of whatever scam Vegas book I read it in.
But, I was reminded of the Martingale on my last trip to Vegas. As I sat waiting for a friend to finish his bungee jump over the strip, I happened to have time to kill at a Blackjack machine that was interesting. Not only did it have a minimum bet limit of $0.25, doing this strategy at a slot machines have perks. Like letting you get in hands more rapidly as you don't have to wait for anyone else at the table. And the massive bonus of no human dealer ready to shame you and your stupid strategy.
On top of these benefits, this machine had another little bonus that made the Martingale strategy sound even better. In this particular machine, after you lost a hand, you are presented with an option to "double down" your bet on the chance of a virtual "coin flip". If you won, you'd recoup the loss and be back to even. If you lost, you just had to double down that bet on the next hand of virtual Blackjack.
Now, in full disclosure, the idea to try this strategy came to me as a vodka induced "epiphany" that always saying yes to a coin flip question would slighly increase my odds from 1% handicap to somewhere closer to the 50% odds of guessing a coin flip. The fact it asked you to double a bet only after losses was another reminder of the Martingale. [NOTE: this obviously carries the assumption that whatever regulatory gaming commissions overseeing the code on machine gaming would likely not allow a gambling machines to present a 50/50 chance of a fair coin, and behind the scenes be at the core programmed as anything different.]
And on top of this, somewhere along the line I had a "brilliant" addition I made while testing the Martingale strategy - fueled by my strong willpower! If I could lean on this incredible gift of willpower by promising myself that after every winning hand, I would simply place my profit into a bucket. And this bucket was never to be gambled.
I noticed immediately the introduction of a coin flip was not just giving me the 1% advantage over another hand of Blackjack, I was seeing far more profits in an hour than I had before. The coin flip was done in a couple seconds, vs. around the 60 seconds a hand of of blackjack could take. This meant I could get in twice as many hands per minute. So a $1 bet had previously capped my profits to $60/hr at 1 minute per hand, 50% or the time a "hand" was down to 5 seconds or so. This meant I could fit in around 100 bets in an hour... (and $100/hr is apparently the amount of profit that is psychologically pleasing enough to overcome any downsides/risks for me.).
In my time at this coin flip machine, I this increase in profits, not knowing the real "odds" behind streaks kept my bets at $1, as I could mathematically cover any losing streaks I had ever personally had. (FYI - In my ~4 hour sessions across my handful of trips to Vegas, I've probably played 5000+ hands of Blackjack. But in a sessions of only around 300 hands, I have never once seen a streak of losses longer than 9. They are usually corrected after 4-5 with an occasional rare 8 in a row that always seems to do the trick).
But if I banked all my profits, I never really risked more than the amount of money I had already come willing to gamble. But even if I lost all of my gambled money, any of the profits I made during the time playing was still there. I had removed from the equation. Can't lose what you never bet. As I applied this plan and the modified Martingale to my new favorite machine, I never lost my money. And I never left without a few hundred in profit. 100% of the time. And all that limited my profits was two simple factors. How.l much each bet was and how many bets you can actually fit into whatever time you have to a play.
So. I have to ask... Am a missing something in the math here? Is my consistent luck with this just that? A mathematical fluke? Is there any way besides simulations to know how often you should see a streak occur in a set amount of time? I am very curious to min/max my bet amounts to a acceptable risk level over a realistic playing time, because I know I could comfortably risk larger bets simply due to my experience of the low number of long streaks I've personally seen and my willpower to never tap into the profits I make.
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Martingale Roulette system - why it doesn't work Martingale Betting System Martingale Strategies don’t work. Or do they? Why The Martingale Betting System Doesn't Work TAI LOPEZ YOUTUBER CONSPIRACY THEORIES Part 1

The Martingale System is a betting strategy that originated in 18th century France and remains popular today. The principle is simple: each time you lose a bet, you double your next bet, so that the eventual win leaves you with a small profit equal to your original stake. The Martingale Betting System. The Martingale Betting System is one of the most popular betting systems used in Baccarat and also one of the oldest in the world.. This is a so-called negative progression system that you can use when you play Baccarat and a variety of other games like Roulette and Craps.. When you play Baccarat, the Martingale betting system works as follows: To every trader own way of trading; I was expressing my observation and experience based opinion. Too strongly, perhaps. I did not mean to be judgmental towards anybody, and apologise for coming across as if I was. The Martingale player would play for 100 bets, or until he couldn’t cover the amount of a bet. In that case, he would stop playing and leave with the money he had left. In the event his 100th bet was a loss, he would keep betting until he either won a bet or couldn’t cover the next bet. The person flat betting would play 100 bets every time. The theory behind the Martingale betting system is a win will eventually follow a loss at some point. For example, you place a wager on the horse with the best odds in a racing tournament.

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