The Gambler's Fallacy re Miracle

TheMadFool

The gambler's fallacy arises from mistakenly assuming that two unrelated chance events have some sort of causal connection. In concrete terms this fallacy is committed when a gambler predicts a head on a coin-flip upon seeing a string of tails in a game. The gambler mistakenly thinks that the past event (series of tails) affects future events (a head is in order to even things out). The truth of the matter is that the probability of getting a head/tail is always 50%. A series of heads doesn't increase the chance of getting a tails and vice versa. All of this provided the coin is fair i.e. both heads and tails are equally likely and the coin is unbiased.

Now imagine someone hands you a coin and asks you to find out whether it's fair or biased. What do you do? Here I introduce two concepts viz. theorertical probability and experimental probability. Here, without going into the details of the two concepts, it should suffice to say that theoretical probability is purely abstract and experimental probability requires the coin to be actually flipped. Thankfully in the case of flipping coins both theoretical and experimental probabilities can be easily calculated. If the experimental probability deviates from the theoretical probability by a significant amount (either too many heads or too many tails than expected) we are justified to suspect the coin is biased/loaded.

So, when we see a series of heads/tails in a coin-flip it increases the chance that the coin is biased. Said in a different way, but staying with the truth, if a coin is unbiased we shouldn't see an inordinate number of heads/tails in a row.

Does this not show that the Gambler's fallacy is not a black-or-white thing(for lack of a better word). Rather it is a matter of degree - the greater the streak of heads in a row the greater the probability that the coin is biased. If the coin is fair then it is reasonable to expect/predict that the next flip will yield a tail. So, the Gambler's fallacy is NOT always a fallacy. It is a fallacy when the numbers are small but when experiments/observations are done large scale it makes sense.

If you're with me on this then we come to the issue of miracles. Miracles are astonishing because they defy our understanding of our world - its behavior, its rules and principles. They're rare occurrences or mathematically, the probability of a miracle is extremely low. However, they have occured in the past - Jesus walked on water and rose from the dead etc. That's 2000 years ago. So, at the risk of committing the gambler's fallacy I predict that another major miracle is due on earth. I have no idea what shape and form this miracle will take but it has the power to completely transform our lives. Imagine what another Jesus-like miracle will achieve?

tom

Thankfully in the case of flipping coins both theoretical and experimental probabilities can be easily calculated. If the experimental probability deviates from the theoretical probability by a significant amount (either too many heads or too many tails than expected) we are justified to suspect the coin is biased/loaded. — TheMadFool

You are only justified in thinking the coin is PROBABLY loaded. As you know, there is no such thing as a finite sequence of tosses incompatible with either fairness or bias.

TheMadFool

You are only justified in thinking the coin is PROBABLY loaded. As you know, there is no such thing as a finite sequence of tosses incompatible with either fairness or bias. — tom

So, in simple terms:

If you get a series of heads then the coin is probably loaded

In other words:

If the coin is fair then we shouldn't get a series of heads

As the series of heads increases in number the probability the coin is biased increases or if the coin is fair then we should expect a tails.

The gambler's fallacy is not completely fallacious.

A miraculous event is due in earth's history.

tom

↪TheMadFool

Jesus didn't walk on water or rise from the dead.

Chany

Your example of someone asking about a biased coin actually removes the gambler's fallacy. The fallacy occurs when people assume that statistics will even out in the near future or that because one thing has happened a lot, the other option must occur. The gambler's fallacy only occurs when the original set-up is fair or that there is no causal reason to create an unexpected spread. If we do a bunch of large scale tests and find that a particular coin is biased, then believing the coin is biased towards one side or the other is not fallacious; it is using experimentation and probability analysis to find a statistically significant outcome.

In other words, you commit the gambler's fallacy when, based on a small sample of an unlikely or repeated outcome, you believe the universe will equal out the probability in the immediate future. If you run multiple, largely-sized tests and find that a particular outcome keeps on occurring in a statistically significant manner, you are establishing a causal link. The only wiggle room is exactly where the line is to be drawn between small and large sample sizes, but we have statistical methods, given a particular confidence level and confidence internal, that should tell us how many samples we should need. Of course, there is always the problem of getting an incredibly unlikely occurrence of error, but there is always the possibility of error in practically all things, so bringing that up does not create problems.

Even if I accept the idea there is a large grey area between small and large sample sizes, you still might be committing the fallacy within the questionable area and are definitely doing so within the small sample size region. You are problematic in multiple areas. You assume miracles (not just statistically unlikely events, but a disruption of the natural order) actually have occurred. You commit the gambler's fallacy by saying we are due for events to happen if we assume so statistical occurrence of miracles. The entire argument you present is effectively a diversion: the line of reasoning you argue for does not even apply in the case of miracles. Even if we believe miracles have occurred, we have no reason to believe they occur again at all, especially with any frequency.

TheMadFool

↪tom

Maybe I'm wrong in the details but I'm quite sure he performed some miracles.

Terrapin Station

Although it seems to me that if the probability of getting heads or tails is really 50%, then if we have a bunch of one side in a row, that should increase the odds of getting the other side on a subsequent throw. Why am I thinking this? Well, for the 50% to have any real significance, it needs to be referring to what happens over a series of throws, where the more throws there are, the closer the data set gets to 50% for either side. Otherwise, how in the world would we be arriving at the 50% figure in th first place?

TheMadFool

You are problematic in three areas. You assume miracles (not just statistically unlikely events, but a disruption of the natural order) actually have occurred. You commit the gambler's fallacy by saying we are do for events to happen if we assume so statistical occurrence of miracles. The entire argument you present is effectively a diversion: the line of reasoning you argue for does not even apply in the case of miracles. Even if we believe miracles have occurred, we have no reason to believe they occur again at all, especially with any frequency. — Chany

Firstly I'm happy that you more or less agree with me regarding the nature of the gambler's fallacy - that it isn't fallacious over large observational data showing biased data points.

Actually there's a philosopher or mathematician (i forgot his name) who's proven mathematically that miracles are quite commonplace events. Of course his main thrust was that for that reason alone we needn't look for a supernatural explanation for miracles. So, I don't quite understand what you mean by ''You assume miracles (not just statistically unlikely events, but a disruption of the natural order) actually have occurred'' and ''You commit the gambler's fallacy by saying we are do for events to happen if we assume so statistical occurrence of miracles.''

You seem to be assigning a probability value of zero to miracles and that basically means certainty that such events can NEVER occur. However the problem of induction should coax/coerce you to deny yourself such privileges. Miracles do have a non-zero probability somewhere between 0 and 1. Say you assign it a value of 1 in a million (1/1,000,000). Suppose an event occurs every second. In 1 minute you have 60 events. In 1 hour you have 60×60 events. In one day you have 60×60×24 events. In one month you have 60×60×24×30 events. In 1 year you have 60×60×24×30×12 events (31,104,000 exactly). That means in 1 year we should be experiencing at least 31 miraculous events [(1/1,000,000) × 31,104,000]. You could of course assign any probability to miracles 1 in a billion or 1 in a trillion; it doesn't matter because the sheer volume of events occurring in the universe will mean that miracles are actualized. The above argument was offered by the philosopher/mathematician whose name I forgot.

TheMadFool

↪Terrapin Station

Theoretical probability of event A occurring = (number of favorable outcomes) ÷ (total number of possible outcomes)

In the case of a coin:

Favorable outcomes for a heads = 1 (there's only 1 head)

Total number of outcomes = 2 (head and tail)

So, theoretical probability of a head = 1 ÷ 2 = 50%

tom

So, theoretical probability of a head = 1 ÷ 2 = 50% — TheMadFool

Unless the coin, or some aspect of its flipping, is biased.

Chany

Firstly I'm happy that you more or less agree with me regarding the nature of the gambler's fallacy - that it isn't fallacious over large observational data showing biased data points. — TheMadFool

I do disagree. The gambler's fallacy is a fallacy in thought: you either committed a fallacy or you did not. If you believe that the tenth dice roll will be a six because the nine previous rolls did not have a six yet, you have committed the gambler's fallacy. If you run three three experiments with a sample size of 1000 showing with the dice is biased against rolling a six, then you are not committing any fallacy, but showing a causal connection. You are not doing anything remotely like the gambler's fallacy: making an unwarranted statement based off a belief that theoretical statistics equal out in practice, especially in the near future of a small sample size.

It's akin to saying that the straw-man fallacy is not always fallacious because, sometimes, people actually engage their opponent's arguments. The straw-man is a fallacy. You either engage in it or you do not. The fact that some people engage actual arguments does not validate the straw-man fallacy or show that it is not always fallacious; they are no longer committing a straw-man fallacy.

Simply put, your argument is nonsensical. Based off of other arguments you put forth on the site, this is just another case of you wanting a conclusion to be true and trying to dress up your justification with poor logic.

TheMadFool

↪Chany

Littlewood's Law

Chany

↪TheMadFool

Littlewood's law refers to the type of "miracles" of unlikely chances. It is meant to explain why things that are unlikely are not really miraculous as people think. It is not really a proof or a law per se, but more of a rule-of-thumb thought experiment to illustrate a point: when someone survives an unlikely scenario or something really wonderful occurs, these are not really special, but to be expected. We noticed the woman who survived falling while skydiving out of a plane with a defective parachute. We notice this extremely unlikely event and might call it "miraculous", but, given probability, there would eventually be a person who survives a fall at an extreme height. So, while from the perspective of the surviving individual and others around them, the event is "miraculous," the individual just happened to be the inevitable one is a million, so to speak.

Jesus' miracles are not just unlikely chances- they are things that defy the laws of nature. Their explanation is divine intervention from an all-powerful and all-knowing being. If their origin is literally naturally impossible, you could never apply statistics to the situation.

TheMadFool

Littlewood's law refers to the type of "miracles" of unlikely chances — Chany

I agree. Littlewood's law limits itself within the boundaries of what we call natural. Taking your example of the sky diver surviving a fall the law would still look for natural causes for the unlikely event e.g. strong updrafts in the air that breaks the fall or falling into water, etc.

When it comes to Jesus-like miracles Littlewood's law seems inapplicable. Nevertheless the problem of induction - the fact that we can never be sure that natural laws will hold for ALL observations - allows us to assign a non-zero probability to Jesus-like (suspension of natural laws) miracles.

If any event has a non-zero probability then Littlewood's law becomes applicable. Therefore, Jesus-like miracles can occur given the sheer volume of events that can occur over prolonged periods of time.

TheMadFool

Unless the coin, or some aspect of its flipping, is biased. — tom

Yes

SophistiCat

Although it seems to me that if the probability of getting heads or tails is really 50%, then if we have a bunch of one side in a row, that should increase the odds of getting the other side on a subsequent throw. Why am I thinking this? Well, for the 50% to have any real significance, it needs to be referring to what happens over a series of throws, where the more throws there are, the closer the data set gets to 50% for either side. Otherwise, how in the world would we be arriving at the 50% figure in th first place? — Terrapin Station

Probabilities are single-case, or nothing

But even if you do not agree with that view, you are still committing a fallacy in attributing a causal significance to that 50% probability. How did we arrive at the 50% figure in the first place? From the following assumptions:

There are two possible outcomes,
Trials are independent,
On each trial there is no more reason to expect one outcome than the other.

Assuming a causal influence of preceding trials on subsequent trials would go directly against those assumptions.

TheMadFool

Assuming a causal influence of preceding trials on subsequent trials would go directly against those assumptions. — SophistiCat

Perhaps the logic doesn't involve causality as is commonly understood.

I do believe that preceding trials and subsequent trials are independent of each other i.e. there's no causal connection.

However, taking the coin-flip example, the appearance of an inordinate number of heads or tails does raise suspicions on the fairness of the coin.

Expressed alternatively, a fair coin should not result in inordinate sequences of heads/tails.

This gives us a valid reason to conclude that in the event of a long sequence of heads a tail is due.

SophistiCat

↪TheMadFool

You were doing well until the last paragraph. Not only does it not follow from the reasoning that preceded it, it goes directly against it! If you concluded that the coin was biased towards heads, then you should bet on heads, not on tails.

TheMadFool

↪SophistiCat

Sorry to disappoint you. You seem to understand better than me. Let me make my reasoning explicit (using the coin scenario)

If you see a an inordinate number of heads then it is likely to be biased.

Taking the contrapositive of the above conditional we get:

If the coin is fair then you shouldn't see an inordinate number of heads

The coin is fair (my assumption)

Therefore, you shouldn't see an inordinate number of heads.

If you shouldn't see an inordinate number of heads then it is reasonable to expect a tail to even out the outcomes.

So, it is reasonable to expect a tail to even out the outcomes

Chany

↪TheMadFool

You then didn't understand Littlewood's law, which arbitrarily defines miracles as one in a million and just meant to show how we should not be surprised when statistically unlikely events occur. Jesus' miracles, even if they can be assigned statistical numbers, would be so astronomically large to occur that we should not expect to see them. More importantly, we cannot assign statistically value because the occurrence of the event is entirely dependent on an all-powerful and all-knowing being who we have no reason to believe will ever create another miracle. In short, we have good reason to believe that Jesus' like miracles are not based on statistics, but careful planning by God.

Notice how we shifted from trying to validate the gambler's fallacy to defending something else entirely. I'm sure the second your interpretation of Littlewood's law is shown faulty, you will shift to some other topic or back to defending your erroneous view on fallacies.

If you shouldn't see an inordinate number of heads then it is reasonable to expect a tail to even out the outcomes.

So, it is reasonable to expect a tail to even out the outcomes — TheMadFool

This is the gambler's fallacy. You believe the 50/50 chance ratio is some magical causal power that the universe must equal out. It is not. A single coin flip is 50/50 heads or tails. Every coin flip has the same chance of being either heads or tails. Previous coin flips have no causal bearing on the outcome of the next coin flip. Therefore, if we flip a fair coin and it lands heads 100 times prior, the likelihood of the 101st coin flip landing heads or tails is still 50 percent each. The reason it seems odd to us is we believe statistically unlikely things will not happen, so the next coin flip will prevent an even less likely event from occurring (Littlewood's law and various other thought experiments explain why this is not the case).

TheMadFool

Jesus' miracles, even if they can be assigned statistical numbers, would be so astronomically large to occur that we should not expect to see them. — Chany

Take Jesus-like miracles to be trangressions of natural laws and not simply unlikely events. The problem of induction which science suffers from logically implies that we can't be 100% sure that, say tomorrow or a hundred years from now, these so-called laws will hold true. This logically implies that Jesus-like miracles have a non-zero probability between 0 (impossible) and 1 (certain). The moment a probability value can be assigned to an event Littlewood's law becomes applicable. Say we assign probability of 1 in a trillion or quintillion or any highly unlikely number you fancy. This very improbable event will be actualized given the sheer volume of events that are happening in the universe.

In short, we have good reason to believe that Jesus' like miracles are not based on statistics, but careful planning by God. — Chany

I haven't talked about god at all. Nevertheless the religiously minded folks will likely interpret such events as divine in origin.

Notice how we shifted from trying to validate the gambler's fallacy to defending something else entirely. — Chany

I agree I have shifted focus from the Gambler's fallacy to Littlewood's law but the two are not so disconnected from each other re my POV. Littlewood's law shows miracle are possible and the Gambler's fallacy??? shows that a miracle is due.

This is the gambler's fallacy. You believe the 50/50 chance ratio is some magical causal power that the universe must equal out. It is not. A single coin flip is 50/50 heads or tails. Every coin flip has the same chance of being either heads or tails. Previous coin flips have no causal bearing on the outcome of the next coin flip. Therefore, if we flip a fair coin and it lands heads 100 times prior, the likelihood of the 101st coin flip landing heads or tails is still 50 percent each. — Chany

If you'll read my post to Sophisticat you'll understand what I mean by saying the Gambler's fallacy is NOT a complete fallacy

The Gambler's Fallacy re Miracle

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