112 votes

Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?

Since you have asked for a non-formal answer, I shall try to oblige by not using any numbers or equations. Fundamentally, your question is, how does it come about that individual events can be ...
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42 votes

Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?

If the probability of heads = p , then the probability of tails = 1-p . If it's a fair coin, then p = 1-p and the probability of either heads or tails is p = 1/2. Now suppose the number of coin ...
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42 votes

Isn't the notion that everything will occur in an infinite timeline an example of the gambler's fallacy?

It looks like you've hit upon the concept of almost surely in probability theory. Something occurs "almost surely" if it happens with probability 1, but there still exist situations where that thing ...
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28 votes

Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?

The convergence appears pretty quickly. This is your faulty assumption. It does apear pretty quickly. In most cases. But not at all every time. There are in some sense two layers of likelyhood: In ...
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  • 11.4k
28 votes

Have I found a paradox, or is the universe digital? Or am I just plain wrong?

We run into essentially the same problem almost any time we try to combine the real numbers as described by mathematics with probability theory. When applying probability theory to something like a ...
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  • 381
20 votes

Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?

Yet we also know that the series will converge upon an equilibrium of heads:tails. I think this is your central problem. This is indeed the most probable result of a series of coin tosses, but ...
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  • 309
20 votes

What is unique about the quantum state of superposition?

I recently answered a similar question on physics.SE here. What is special about the probabilities of quantum mechanics is that the randomness cannot be explained by a theory of nature that is both ...
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17 votes

Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?

If you use a fair coin, the average of heads thrown will converge to 50%. However, the number of heads won't converge to half the coins thrown. While the percentage comes closer and closer to 50%, ...
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16 votes

Is there any rigorous definition of just one single random choice?

It's best to think of randomness as a model, not as a 'thing'. When we talk about a 'random event', we mean that at some time t0 we cannot predict with certainty the state of a system at time t1 (t1 &...
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15 votes
Accepted

Isn't the notion that everything will occur in an infinite timeline an example of the gambler's fallacy?

Here, I think, is a more succinct answer: Let's say we have a dice with 1 trillion sides. Then, the probability of a given outcome on the next roll of the dice is one-in-a-trillion. On the other ...
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13 votes

Have I found a paradox, or is the universe digital? Or am I just plain wrong?

Zeno's Paradox is not a paradox. It is an attack on loose thinking. By emphasising the infinite nature of one thing, and not mentioning the infinite nature of another, it confuses people into thinking ...
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12 votes

Why are homologies evidence for evolution instead of common design?

Affirming the consequent Scientists distinguish between the merit of explanations on the basis of (a) how accurately and (b) how widely they make experimentally-verified predictions. This means ...
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  • 506
12 votes

Is getting 100 Heads in a row from a fair coin a miracle or not?

Yes, but only because every possible sequence of flips is a miracle. Consider. Suppose I flip a fair coin 10 times in a row. Say I get HTTHTTHTHT. Now that seems pretty normal, nothing remarkable ...
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  • 2,871
11 votes

Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?

This is really math, not philosophy. Assume that you've tossed the coin so far m times and gotten n heads. The fraction of heads so far is n / m. Now you toss the coin one more time. There is a 50% ...
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  • 535
11 votes

Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?

Because "converge to an equilibrium" doesn't mean an exactly equal number of heads and tails, it means the proportion of heads to tails approaches equality (with probability 1: the meaning of which ...
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11 votes

Isn't the notion that everything will occur in an infinite timeline an example of the gambler's fallacy?

"If you made 1,000,000 similar decisions, the probability of that final outcome being reached at any one moment is 1 in a million." That quote represents the root of your misconception. If a coin is ...
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  • 267
11 votes

Is the notion of "Complex System" a philosophy of science? Is it the opposite of Reductionism? Is it related to Holism?

Short Answer Complex systems is a mathematical approach to studying certain objects of science, and is neither a science, nor a philosophy, but an approach that might be considered a combination of ...
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11 votes

Is the notion of "Complex System" a philosophy of science? Is it the opposite of Reductionism? Is it related to Holism?

A system is a complex system if its characteristic properties cannot be investigated by studying its components in isolation. A typical example of a complex system is the weather: One cannot study the ...
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10 votes

What is the difference between a probability and a possibility?

On the prevailing extensional interpretation of modality the difference between possibility and probability is the diffference between quality and quantity, possibility is the quality quantified by ...
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  • 41k
10 votes

Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?

You are comparing two different cases. One is "the probability of landing heads on the next flip" and the other is "sum of the number of heads." The latter is governed by the Central Limit Theorem, ...
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  • 17.3k
10 votes

Have I found a paradox, or is the universe digital? Or am I just plain wrong?

The problem is that probability 0 does not mean 'impossible'. If you have someone flip coins forever, what is the probability that he will never encounter a head? Well, it's zero. But it's possible! ...
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9 votes

Is a "fair coin toss" a logical contradiction?

I'm not sure that I entirely agree with your analysis. Firstly, and perhaps somewhat pedantically, let's note that although randomness has a precise definition in certain mathematical theories - e.g.,...
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  • 3,452
9 votes

Isn't the notion that everything will occur in an infinite timeline an example of the gambler's fallacy?

You're right about the gambler's fallacy, but you're missing something essential about infinity. Infinity doesn't stop. So, you've got your immortal monkey and his endless reams of typewriter ...
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  • 1,204
9 votes

Books and papers on the philosophy of probability

The most self-contained book I am aware of on the subject is Philosophical Theories of Probability by Donald Gillies (Routledge, 2000): I believe reading the historical development of ideas is ...
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  • 1,340
8 votes

Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?

You are right: After a series of 10, 20, 40, 80 heads the probability for another head is still 1/2. It is not slightly less or slightly bigger, it is constantly 1/2. Tosses have no memory. To ...
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  • 20.6k
8 votes

Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?

To build on what celtschk pointed out (and possibly others, I haven't read all of them) with more examples, 'tend towards 50/50' is not something as in the next n throws will negate any off-set that ...
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  • 221
8 votes
Accepted

Hilbert's Sixth Problem: Is Kolmogorov's solution the last word?

There is no general agreement on the axiomatisation of probability. Kolmogorov was a frequentist and his approach proceeds by supposing the existence of an event space, or possibility space, defining ...
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  • 15.7k
8 votes

Is there a name for the fallacy: 1/100 chance == 100 times guaranteed success?

This is definitely closest to the gambler's fallacy. An example of this fallacy demonstrated in your example would be that the player, having killed 80 monsters without a coin, thinks that the next 20 ...
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7 votes

Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?

You need to be careful to specify the question you are asking. Going forward, the coin has no memory and the chance of heads on any given toss is 1/2. Period. End. The convergence to the mean is ...
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