Help me convince friend that Martingale Betting Strategy Doesn't work

[bos]

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I play 8 hours per day, 7 days a week. I bet and get a result on average once every 2 minutes, so 30 bets on average each hour, and 240 bets each day. There is NO limit on the table, and I'm guaranteed action from the casino.


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Well, you'd make $1 about every 2 bets until you finally busted out. This works out to $120 per day.

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How long could I play before I should expect to go broke?


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Well, if you play with american roulette red or black bets where the casino wins 52.6% of the time, then you will on average survive for:

Chance of busting = (0.526 ^ log2(bankroll))
Avg wins before busting = (1 / chance of busting)

wins before you finally bust (don't have enough $ to make the recovery martingale bet).

You can plug in the house's chance of winning for any other game into the formula instead of .526.

For a $1000 bankroll, you would survive for, on average, 603 wins. With the rates you gave, it would take you on average 5 days before you hit the losing streak that busts you. Of course, it could happen the first time you play.

There is actually a fairly small chance that you will double your money (we'll discuss that later) before you bust. If that happens, you will usually have one more "chance" before you bust. If you pass a milestone, like $1024 (2^10) or $2048 (2^11), then your average time until busting improves.

The above formula does not take this bonus into account when estimating average time until bust, though as you'll see, the probability of doubling your money before busting is small. To find your chance of doubling your money before busting out, do this:

(1 - Chance of Busting) ^ Bankroll

With a $1,024 dollar bankroll and $1 bets, you have a 19.7% chance of doubling your money before you bust.

With larger bankrolls, your chances of doubling drop. With a $1,000,000 bankroll your chance of doubling before you bust is about 6.4%.

With flat $1 bets and no martingaling, you would last on average for 38 million bets before you finally busted out. Your odds of doubling your money, however, with $1 flat-bets are practically nil.

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I guess that would be some type of bell curve or something. 20% chance of going broke in X days, or 80% chance of going broke in X days, etc. Also, how much would I make per day typically when I didn't go broke? Perhaps the same question with a $5 base bet.


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When you don't go broke, you make $1 for about every 2.05 bets you make. With a $5 base bet, you make money 5x as fast when you don't bust. Just do all the computations for number of wins before busting with your bankroll divided by 5. You will go broke in far fewer bets. Your odds of doubling, however, are much increased.

Your best chance of doubling, however, is to bet the entire bankroll at once.

If there were no house edge, your chance of successfully doubling is exactly the same whether you martingale or bet it all at once, or make flat bets for that matter.

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I honestly have no idea what the results will be, but if they come out a certain way, I will loudly proclaim that Martingale can't work in a practical setting (because I think my question IS a practical one), and that my instinct on this is just plain wrong.

I'm really quite curious. What the heck would happen in a scenario (or similar) to the one above? Anyone feel like doing the math, since I certainly can't?

Thanks.

Sparks

Edit: I realize putting the money in a bank account of some sort would be the smart thing to do, so just assume I love to gamble, or don't trust banks, or something :-)

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[Sparks]

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I honestly have no idea what the results will be, but if they come out a certain way, I will loudly proclaim that Martingale can't work in a practical setting (because I think my question IS a practical one), and that my instinct on this is just plain wrong.

I'm really quite curious. What the heck would happen in a scenario (or similar) to the one above? Anyone feel like doing the math, since I certainly can't?

Thanks.

Sparks

Edit: I realize putting the money in a bank account of some sort would be the smart thing to do, so just assume I love to gamble, or don't trust banks, or something :-)

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Thanks Bos.

So if I play craps, which has slightly better odds (51.5% house edge?), and use a base bet of $2, I can average $240 per day. With a $4M bankroll, I can make $30/hour off the casino, 5 days a week, and not expect to go broke for 18 years.

I suggest by way of the above example that Martingale has real applications, which the casinos recognize, and thwart by imposing wager limits.

Sparks

[bos]

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Thanks Bos.

So if I play craps, which has slightly better odds (51.5% house edge?), and use a base bet of $2, I can average $240 per day. With a $4M bankroll, I can make $30/hour off the casino, 5 days a week, and not expect to go broke for 18 years.

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Yes. 50% of the time, you will bust out in fewer than 18 years, and 50% of the time, it will take more than 18 years to bust out. On average, you will accrue +1.6M before busting. You will make a $128,000 bet about twice a year, 256k bets about once a year, 512k bets every 2 years, 1M bets every 4.5 years, 2M bets every 9 years. And half of those 2M bets, you will lose.

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I suggest by way of the above example that Martingale has real applications, which the casinos recognize, and thwart by imposing wager limits.


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As far as how it looks from the casino's perspective, it's not so bad. A casino with one player would have to wait on average 18 years for the big payoff and is stiffed if the player stops. But, when MANY people play this strategy, the casino's income stream is much more stable. With 180 people doing this, they will bust on average 10 people each year for their full 4 million bankroll. That's worth +$40 million. They pay out $90,000 each to the other 170, for -15.3 million. That's still a gross profit of $25M / year for the casino.

With 1000 players, it's even better. Sign me up, as the casino!

I would agree with you for absolutely enormous stacks (in multibillions) that might dwarf the casino's bankroll, they should be concerned. Whether or not the player wins the once per multicentury billion dollar bet can make a big difference in the casino's business, and there aren't enough players with that kind of bankroll to average it all out. In general, I think casinos want to avoid huge bets, not so much because the player's strategy is any threat, but because by their very nature, they add casino-level variance. I don't think they'd be too keen to let someone sit down with a full billion and put the entire sum on the pass line either.

[Sparks]

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I suggest by way of the above example that Martingale has real applications, which the casinos recognize, and thwart by imposing wager limits.

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As far as how it looks from the casino's perspective, it's not so bad.
(snip)
Sign me up, as the casino!

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Nice perspective Bos. I always thought the ONLY reason there were table limits was to combate Martingale-type systems. From what you've said, I recognize this is probably not the case.

Sparks