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, ...


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 ...


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 ...


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 ...


6

There are plenty of issues with subjective probability assignments to degrees of belief discussed e.g. in SEP's Subjective Probability Theory. I will only address the one outlined in the OP. For a book length treatment see Fundamental Uncertainty: Rationality and Plausible Reasoning volume edited by Marzetti and Brandolini. The idea of distinguishing ...


4

This is called Bertrand's box paradox. It's a "paradox" in the sense that the correct answer, P(other_coin_is_gold) = 2/3, is unintuitive to most people at first. To get a feel for the correct answer, break it down into cases. Let's call the box with 2 gold coins "G", the box with two silver coins "S", and the box with one of each "M" (for "mixed"). Further,...


3

Let me point out, first, that Occam's Razor is not a law; it's a rule-of-thumb that has more to do with pragmatics and aesthetics than necessity. It's a good rule of thumb, sure, but it is based on an a priori belief that the universe as a whole conforms to what human minds count as parsimony. That being said, the main issue with applying Occam's Razor to ...


3

In colloquial terms, conditional probability tries to capture the idea that context matters. For instance, if we ask how likely it is we will be robbed in the coming year, we can do the math: I've seen numbers like a 0.15% chance that any given person will be robbed in any given year. But obviously that's not the same for everyone, everywhere. Someone who ...


3

Wikipedia may be a good place to go for initial information on this. Here is what it says: In probability theory, conditional probability is a measure of the probability of an event occurring given that another event has occurred. This tells us that there are two events involved. We want to find a measure for the probability of one event, A, given that ...


2

Probability is something that must be considered at a specific point in time, otherwise it would have no purpose. Considering this Universe is based on duality, of balance of energy (because it otherwise could not maintain its integrity), it can be safely said that the default probability for something unknown is 50%. Therefore, just not know the ...


2

Your setup and experiment are analogous to the following more general scenario: Suppose you want to provide evidence for the claim that all As are Bs. To do so, you design an experiment that only ever looks at Bs, and willfully ignores anything that isn't a B. If you find a B that isn't an A, no big deal; this doesn't contradict your hypothesis that all As ...


2

This isn't a full answer, but I'd like to point out that you've formulated an alternate version of Zeno's Paradox. As the amount of time increases, the probability that some rare event does not occur becomes smaller and smaller but is never exactly zero. This is similar to how Zeno moves ever closer to but never reaches the target destination. Nonetheless, ...


2

What the issues are, and what "effectively solving" them means is largely in the eye of the beholder, people decide among interpretations largely based on their personal core beliefs about realism, determinism, the role of science and the like. Quantum Bayesianism, with its mix of realism about physics with anti-realism about the structure of quantum theory, ...


1

The short answer is that you can't tell, because you are working from a single data point. For all you know, the dealer has dealt hundreds of thousands of hands and you just happened along at a propitious moment. Royal flushes are rare, but they do happen, even with properly shuffled decks. Winning the lottery ten times in a row is different because you have ...


1

There is a field that adresses your concern, and it's called Management. Scientific management to be more precise. It involves collecting data to make statistics and then make rational decisions. It also involves modelling and testing prototypes in order to get the closest approximation we can get to the real world/full scale situation. Under how much ...


1

One fallacy that is evident in your question but has not been addressed by the other answers is: everything will occur in an infinite timeline And you said something that is an instance of the fallacy: if the Universe is infinite, there must be a planet exactly like ours somewhere Both of these are completely fallacious. Nothing about an infinite process ...


1

In addition to @AdamSharpe's analysis of "break it down into [six] cases", you can also analyze the situation your intuitive way, i.e., first choose a box, and then choose a coin from that box. Start with all three of your boxes: let's call them GG, SS and GS, with the obvious meanings. Then, first of all, if you happen to choose the SS box, then your ...


1

(I'm guessing you're learning about this in the context of Bayesian epistemology, right? "Credence" plays the roll that a probability assignment does in regular probability theory. I'll answer the question using the common probability theory notation. It should be an easy but useful exercise for you to then translate the concepts into your "credence" ...


1

So, there are three models of quantum mechanics which maintain determinism: non-local hidden variables; superdeterminism; and determinism across many worlds. These are constrained by the Bell Inequality having being violated. This is because, sensitivity to unitial conditions, the underlying characterustic of chaotic systems (such as whenever more than two ...


1

As noted in the comments, tracking the real sources of randomness is controversial, and depends on metaphysical views about determinism. As we are ignorant of the "true metaphysics", one could say that our world itself is (for now) just such a system. Instead, I'll give a mathematical example. An accessible exposition of the related issues is a recent survey ...


1

I suspect the underlying problem you are concerned about is whether it's justifiable to assign probabilities to claims with insufficient evidence, and if so, how can such probabilities be assigned? For some responses to Pascal's Wager, this comes in the form of assigning a probability to the existence of God. Often, it is justifiable to assign probabilities....


1

I'm not sure what you mean by people "arguing in favour of IQ"; perhaps you mean people advocate specific policies because they seem liable to increase cognitive abilities, such as improved education of removing lead from the environment. OK, I grant people do that: we want to give people as much of whatever IQ tests measure as we can, but also want them to ...


1

As to Heaven, I don't expect anyone to be able to validly infer the laws of any non-physical world, real or imaginary, from the laws of the physical world. Why would time in Heaven necessarily be physical time? Why expect a transition from the physical to the non-physical to follow the laws of the physical? If heat death, particle decay and all the laws ...


1

An interesting point about your question is that Occam first applied the Razor - though the term is not his - to a metaphysical issue, the problem of universals. Ockham was a nominalist - he denied the existence of "universals." What are they? Let us begin with what they are not. Universals contast with particulars. Particulars are the individual things ...


1

In short, propensities are, precisely, the quantum mechanical probabilities. In a classical, deterministic, world we set up approximations to the ideal of a repeatable experiment being understood that each experimental run differs from the others in small mechanical variations. In a classical context propensities are thus extracted from the deterministic ...


1

The basic problem with the OP is that the first numberic value of 98 percent or whatever already takes into account all possible assumptions. I am not saying the number is correct or whether arriving at any such value is even possible. But if it is at all possible. It has to be done in the first step itself. There is no need or possibility for further ...


1

I must point out that, even if one starts with a biased coin (in the sense that the probability of tossing heads is always P, tails always 1-P, but these are not necessarily equal), it is always possible to synthesize a toss for which heads and tails have equal probability. The procedure is simple: Toss the coin twice in succession; if the two tosses show ...


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