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 ...
- 22.8k
43
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 ...
41
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 ...
- 1,177
37
votes
Accepted
Why would infinite monkeys not produce the works of Shakespeare?
Yes, the monkeys will do it. No, they don't have to.
It's mathematically true that after removing all logistical constraints - which is what we mean when we say there are infinitely many monkeys, ...
- 620
33
votes
Is the SETI project built on false premises?
Scientists use statistical significance measures all the time, discussed here: How improbable does an event have to be before we can say it didn't happen by chance?
"Life occurring on earth ...
- 20.5k
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 ...
- 13.5k
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 ...
- 381
23
votes
How does probability constitute as knowledge in justified belief theory?
There is no paradox here. What you knew for certain before the lottery was that there were 100,000 possible outcomes, in only one of which you would win. The result of the lottery does not change the ...
- 13.1k
23
votes
Does Bayesianism give an out for pseudoscience that it shouldn’t deserve?
The first issue is that the presented hypothesis is actually two hypotheses. While the sentence "Adam can guess, using psychic powers, what the price of each stock in the world will be at the end ...
- 359
21
votes
Accepted
Does every possible event have non-zero probability?
The answer is no. Mathematically, if you have a continuous random variable, the probability of getting any one of its values is zero, but you can still get one, so zero probability does not ...
- 13.1k
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 ...
- 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 ...
- 484
19
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%, ...
- 5,243
19
votes
Accepted
Is all of probability fundamentally subjective and unneeded as a term outright?
In fact there is no way to define unique objective probabilities for anything that happens (or not) in the real world. Every definition of probability requires model assumptions that cannot be ...
- 573
19
votes
Isn’t everything absurdly improbable?
The probability of getting 100 straight heads is the same as any other sequence, yet the other sequences aren’t seen as improbable.
Yes they are, if you put a piece of paper in a sealed envelope ...
- 1,730
17
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 ...
- 326
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 &...
- 18.5k
16
votes
Isn’t everything absurdly improbable?
Isn’t every event by definition improbable in the sense that each
event precedes an infinite series of causes that could have
(theoretically atleast) been different?
Specific events don't have ...
- 3,400
15
votes
Does every possible event have non-zero probability?
In Bayesian theory, the principle that every logically possible proposition has a probability that is never exactly zero, nor exactly one, is called the Cromwell Rule.
The idea behind the principle is ...
- 22.8k
15
votes
Accepted
Is it ever rational to stumble onto the conjunction fallacy in probability?
Well, the conjunction fallacy is a fallacy, so it's not rational to believe it. However...
Suppose Dave tells you he was just robbed and that's why he's were late to meet with you. You are skeptical, ...
- 11.1k
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 ...
- 139
13
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 ...
- 2,890
13
votes
Should X, if there’s no evidence for X, be given a non zero probability?
You mention Bayesians, so I'll reply in that context.
The Bayesian framework is (roughly) a way to take a "prior probability" of a statement, multiply it by a factor accounting for new ...
- 1,214
12
votes
How does probability constitute as knowledge in justified belief theory?
In a lottery, the expectation that you will definitely lose is not justified. The expectation that you are very likely to lose with a tiny chance of a huge win is justified.
Now the fact that you will ...
- 5,243
12
votes
How does probability constitute as knowledge in justified belief theory?
It is irrational to buy a lottery ticket, not simply because the chance of winning are low. The expected value of the money you gain is negative. In other words, if you had enough money to buy all ...
- 2,232
12
votes
Can a coincidence be evidence of a god?
Suppose for example that a person is standing on stage and says “God, if you exist, strike me with lightning right now” and a lightning strike occurs that barely misses him, is this evidence of God? ...
- 1,458
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% ...
- 563
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 ...
- 706
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 ...
- 267
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