# Tag Info

111

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 completely unpredictable but when you pile a lot of them together, either in a sequence or in a mass, the behaviour of the whole pile becomes, if not totally ...

42

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 tosses is N, and let's say that N is getting pretty large. The expected value of the random variable that is the number heads out of the N tosses is going to be ...

41

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 does not occur. The infinite coin flips problem is a great example - with infinite coin flips, you will almost surely see at least one result of heads, that is, ...

28

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 coin flip, a die, or a deck of cards, we use what's known as a Probability Mass Function to assign a probability value to each possible result. What's the ...

25

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 layer one, every single event has the very same probability as its precedessors. In layer two, the sequence of events as a whole has a probability to occur. And ...

19

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 probability doesn't apply to things that are already known to have happened. Imagine this game: A coin is tossed 100 times. Gamblers can bet on the total number ...

16

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%, usually the number of coins will diverge more and more from exactly half. How can this be? Throw ten coins. You'll probably get 3 to 7 heads. 30% to 70%. Throw ...

14

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 hand, the probability of getting a given outcome, at least once, given infinite dice rolls approaches 1. Given enough time, monkeys banging randomly at a ...

13

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 something is impossible. The emphasis, in Zeno's Paradox, is upon the infinite number of times a distance can be subdivided, giving the impression that it ...

12

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 empiricism is fundamentally based on affirming the consequent (and uses inductive reasoning), so you could argue empiricism is rather weak logically. However, you ...

11

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 hides all the mathematical formalism to deal with the possibility of other results). In fact the probability of an exactly equal number of heads and tails after ...

11

"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 tossed 1 million times, the likelihood of any specific sequence of 1 million tosses is 1 in 2^1000000. However, the chances of tossing heads 10 times in a row ...

10

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 probability, see Probability Distributions Over Possible Worlds by Bacchus. This interpretation can be traced back to Leibniz's determinate possible worlds, but ...

10

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, which explains why the sum converges so rapidly (in many cases). Summing acts very differently than simply asking "what's the next result," and its the summing ...

10

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% chance that the toss is tails and the fraction becomes n / (m + 1), and a 50% chance that the toss is heads and the fraction becomes (n + 1) / (m + 1). By ...

10

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! In fact, every specific infinite sequence of heads and tails is infinitely improbable; that is, its probability is zero. Still, none is impossible: one of them ...

9

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., information theory, where we define randomness as the inability to compress information - in a philosophical context randomness has no precise definition. ...

8

This looks like an interesting aspect of the general concept of Moral Luck, which I'm sure you would find worth investigating. The most commonly cited first paper on this stuff is Thomas Nagel's Moral Luck, where he discussed the idea of the Control condition: Without being able to explain exactly why, we feel that the appropriateness of moral assessment ...

8

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 reconcile this result with the naive expectation one should take into account: The probability of a series of length 10 with 10 heads is (1/2) ** 10, which is about 1/...

8

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 is currently in place, it's rather, when n gets big enough any current off-set becomes insignificant. I.e. Let's assume you somehow manage to toss 100 coins ...

8

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 monsters will have a higher probability of finding a coin with each kill. This is fallacious, as no matter how many monsters the player has killed, even after ...

8

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 supplies, and a typewriter with 40 keys. He endlessly hammers on the keys perfectly randomly. The probability that he types a "T" on the first try is 1/40. The ...

7

Whether or not the coin has been tossed, the probability is just a model of how you expect events to turn out, or to have turned out, based on the information you have. Without having information about the result of the coin, it does not matter whether or not it has been flipped yet. The probabilistic model which assigns probabilities based on different ...

7

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 because any excess that you have now will be washed out in much larger numbers. Say the first ten tosses come up heads. At this point, if I asked the most ...

7

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 the probabilities of propositions in terms of their frequency relative to the total size of the space, then defining negation, conjunction and disjunction in ...

6

The comment of Marco have given me another idea of the example of rationality of lottery playing. Let's assume I'm the prisoner on the island, and I'm getting 10\$ a day. 5\$ a day is enough to buy everything I need (food, accomodation etc.) and there's little to be bought on that island. I can't eat twice as much. So the additional 5\$ a day have practically ...

6

Series will generally converge but there is always a small probability that a serie does not converge after a finite number of trials, so there is no contradiction. If you already had 100 tails the whole serie will converge more slowly. The interpretation of probabilities (degree of credence? Objective propensity? Frequency?) is an independent matter.

6

Let's assume you have tossed ten heads, and you are about to do a million further tosses. What is the expectation of the difference between heads and tails? Well, it's ten, because you already have ten tosses, and the expectation for the future tosses is as many heads as tails. Let's for the moment assume that in the next million tosses, you get exactly ...

6

The point is that we should use bayesian reasoning to infer a cause from its consequences. The probability that an hypothesis is true given the evidence is not the same as the probability of the evidence given the hypothesis. For example: the probability that the floor is wet if it has rained is 1 but the probability that it has rained given that the floor ...

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