A Less Obvious Martingale Fallacy - Page 7

Dov · #61 07-17-2005, 05:03 PM

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It's wrong to look at the outcome of a trial (series of trials) and then assigning it a probablity. What you are calculating now is the probability that it will re-occur.. And that probability may well be zero.

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This game has no memory. The probabilities for each trial do not change.

The probability that you will win at least once approaches infinity as long as you have a non zero probability event.

If you are looking at a roulette red/black bet, then using your logic, it would be a leap to assume that no trial would produce a win but we can think of series which would do just that. The problem with this thinking, IMO, is that we simultaneously have the opposite possible series as well, ie: infinite wins.

I think that for the sake of this discussion we can let these 2 series cancel each other out despite the fact that the winning series is actually less probable than the losing one. This is because it doesn't really matter as we are saying that both will occur with a 100% probability.

I say that we can do this, just as we can 'take a limit' and cancel the terms there.

When we take the limit of infinite losses, we also take the limit of infinite wins and they cancel each other out. That only leaves you with series that include at least 1 win for the gambler.

And when he wins, he recoups all of his former losses plus 1 unit of profit.

Since there are no more series that do not include a win, he must win and the casino must lose.

Dov · #62 07-17-2005, 05:06 PM

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I think that for the sake of this discussion we can let these 2 series cancel each other out despite the fact that the winning series is actually less probable than the losing one. This is because it doesn't really matter as we are saying that both will occur with a 100% probability.

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Maybe this is where the problem is.

When we encounter a losing streak, it will be many orders of magnitude larger than its equivalent winning streak.

But when you take an infinite limit, it still doesn't really matter because as was pointed out earlier, infinity + x = infinity.

MMMMMM · #63 07-17-2005, 05:36 PM

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So the gambler must "win" AND the Casino must "win". There will be times when the gambler is ahead and there will be times when the Casino is ahead. So how do you measure who is doing better?

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The casino will NOT win if the gambler is allowed to always complete the series. It's impossible.

If there is a precondition that the gambler is always allowed to complete the series, then the casino must account for its reverse implied odds. They are actually paying out more than they are winning provided that the gambler can always complete his series.

Paradoxically, the only way the casino can win here is to put an end to his play or stop the martingale.

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It might be better if you put the "must" in quotes when stating that the gambler must win a series (because perhaps that is not completely true)

But even if it is true, it is also true the casino "must" eventually win a streak big enough to eclipse all of the gambler's current accumulated one-unit profits. That horrible bad run is actually statistically expected and is probably "inevitable" if the gambler keeps on playing throughout eternity.

So the gambler "must" have times at which he is ahead of the casino. HOWEVER, the casino also "must" have times when it is ahead of the gambler (on the gambler's very worst streaks), and at these times the casino will often be further ahead of the gambler than the gambler ever was ahead of the casino (and all the more so because of the built-in green zero house edge on roulette).

Let's look at this another way, too.

Say there is a large corporation that will never go bankrupt or out of business, but is free to make or lose money. It's profits or losses are somewhat volatile, too, due to its particular niche and structure.

The accountant says, "We will only do the accounting after the quarters or years which show a profit. Quarters or years showing a loss will simply be carried over into the future, until we are showing a net profit, at which time we will do the accounting."

Would that make any sense? (granted it is not a good analogy, but I'm just trying to illustrate the fallacy of selecting your accounting times based on performance).

MMMMMM · #64 07-17-2005, 05:40 PM

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So the gambler must "win" AND the Casino must "win". There will be times when the gambler is ahead and there will be times when the Casino is ahead. So how do you measure who is doing better?

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The casino will NOT win if the gambler is allowed to always complete the series. It's impossible.

If there is a precondition that the gambler is always allowed to complete the series, then the casino must account for its reverse implied odds. They are actually paying out more than they are winning provided that the gambler can always complete his series.

Paradoxically, the only way the casino can win here is to put an end to his play or stop the martingale.

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It might be better if you put the "must" in quotes when stating that the gambler must win a series (because perhaps that is not completely true)

But even if it is true, it is also true the casino "must" eventually win a streak big enough to eclipse all of the gambler's current accumulated one-unit profits. That horrible bad run is actually statistically expected and is probably "inevitable" if the gambler keeps on playing throughout eternity.

So the gambler "must" have times at which he is ahead of the casino. HOWEVER, the casino also "must" have times when it is ahead of the gambler (on the gambler's very worst streaks), and at these times the casino will be further ahead of the gambler than the gambler ever was ahead of the casino (and all the more so because of the built-in green zero house edge on roulette).

These times will become rarer and rarer, but the negative slumps when they do occur will be deeper and deeper for the gambler. So, they both are (nearly) "assured" of being ahead at some points.

Let's look at this another way, too.

Say there is a large corporation that will never go bankrupt or out of business, but is free to make or lose money. It's profits or losses are somewhat volatile, too, due to its particular niche and structure.

The accountant says, "We will only do the accounting after the quarters or years which show a profit. Quarters or years showing a loss will simply be carried over into the future, until we are showing a net profit, at which time we will do the accounting."

Would that make any sense? (granted it is not a good analogy, but I'm just trying to illustrate the fallacy of selecting your accounting times based on performance).

Dov · #65 07-17-2005, 05:48 PM

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HOWEVER, the casino also "must" have times when it is ahead of the gambler (on the gambler's very worst streaks), and at these times the casino will often be further ahead of the gambler than the gambler ever was ahead of the casino

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The only streak that this applies to is one with infinite losses. Any streak that includes a win will be a loss for the casino.

In your business example, it is like offering the clients a full refund on all of their payments to the company if they fulfill a given condition.

You would have to account for the money you would be expected to pay out as a result of clients fulfilling their obligations. This money expected to be paid still needs to be accounted for.

You cannot simply do a balance sheet and get an accurate picture of the business. You would need to do all of the key financial reports and they will include payments expected to be made.

The effect of the zero in the roullette wheel for this situation, as I see it, is that it makes the likelihood of having an infinite losing streak higher than the likelihood of having a winning one.

The problem with this, of course, is that both of those probabilities are 100% and cannot get any higher. This is what I meant when I said that the infinite (unlimited) bankroll would defeat the house edge. It makes both events equally likely.

SheetWise · #66 07-17-2005, 05:58 PM

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But even if it is true, it is also true the casino "must" eventually win a streak big enough to eclipse all of the gambler's current accumulated one-unit profits.

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Disagree - as in a later thread.

PairTheBoard · #67 07-17-2005, 05:59 PM

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So the gambler must "win" AND the Casino must "win". There will be times when the gambler is ahead and there will be times when the Casino is ahead. So how do you measure who is doing better?

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The casino will NOT win if the gambler is allowed to always complete the series. It's impossible.

If there is a precondition that the gambler is always allowed to complete the series, then the casino must account for its reverse implied odds. They are actually paying out more than they are winning provided that the gambler can always complete his series.

Paradoxically, the only way the casino can win here is to put an end to his play or stop the martingale.

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By "win" I meant "be ahead". Sometimes the Casino will be ahead and sometimes the Martingaler will be ahead. If the Martingaler wants to count himself a winner when he's ahead, then he must count himself a loser when he's behind.

PairTheBoard

PairTheBoard · #68 07-17-2005, 06:02 PM

Dov --
"The only streak that this applies to is one with infinite losses. Any streak that includes a win will be a loss for the casino."

You are only looking at the end of a streak. The middle of a streak is just as valid a time to look.

PairTheBoard

MMMMMM · #69 07-17-2005, 06:05 PM

Well, I do not necessarily agree that the probability of having an endless losing streak is ZERO. Or, you could say that I do not necessarily agree that the gambler is CERTAIN to win a future bet.

Anyway, as the chance of a Long/Longer/SuperLong losing streak decreases geometrically (minus the green zero effect), the penalty for having such a streak increases geometrically.

The merely "very long" losing streaks are expected to negate the gambler's accumulated wins, plus some (due to the house zero(s)).

Also, what is the point of taking accounting only after wins, if the gambler is going to always continue playing?

PairTheBoard · #70 07-17-2005, 06:07 PM

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It's wrong to look at the outcome of a trial (series of trials) and then assigning it a probablity. What you are calculating now is the probability that it will re-occur.. And that probability may well be zero.

There are many things we can observe -- then assign a probability -- and conclude that they will never happen again. The leap in logic is to assign a probability of zero to an event you just witnessed.

SheetWise

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There may be many things that are wrong to do, but I don't think there was anything wrong with what I said.

PairTheBoard