4 ms·

Isn't 100 / 100 the most likely outcome for a fair coin? Sure it's unlikely that you hit exactly that result, but every single other result is even less likely.

by fainpul 11mo ago

Isn't 100 / 100 the most likely outcome for a fair coin? Sure it's unlikely that you hit exactly that result, but every single other result is even less likely. What I'm trying to say: if you get 100 / 100, that's not a sign of an unfair coin, it's the strongest sign for a fair coin you can get.

Joker_vD 11mo ago
Well, yes. But the expected deviation from the mean is still ≈7.07. And the probability that the outcome will be either 93/107 or 107/93 is (slightly) higher than the outcome being exactly 100.
- fainpul 11mo ago
  But those are 2 results. 100 / 100 is more likely than 93 / 107 (or any other specific result) is what I'm saying.
  - Joker_vD 11mo ago
    Depends what you count as a result, I guess. "There is exactly N flips of a single kind" is also a viable definition, just as "The exact sequence of flips was x_0, x_1, ... x_199" is.
  - paddleon 11mo ago
    You can tell the difference between 93 heads, 107 tails and 93 tails, 107 heads but not between 100 heads, 100 tails and 100 tails, 100 heads. So while those are two results (93/107 and 107/93), they really only count as two separate outcomes if you pre-specify that the first number is heads. If instead you consider symmetries, where there are 2 ways to get 93/107 and only one wall to get 100/100, then you have more likelihood for the 93/107 outcome because you have two ways to get it.
    - deleted 11mo ago
      [deleted]
  - kalaksi 11mo ago
    I don't think that makes 100 / 100 the most likely result if you flip a coin 200 times. It's not about 100 / 100 vs. another single possible result. It's about 100 / 100 vs. NOT 100 / 100, which includes all other possible results other than 100 / 100.
    - andoando 11mo ago
      If you flip a coin 200 times, and repeat that a billion times, 100/100 will be more likely than any other ratio. It will be a bell curve with 100/100 as the peak
- eviks 11mo ago
  Why not go one abstraction further and go expected deviation from deviation? Probably the word "expected" plays a mind trick? "Expected" doesn't mean the probability increases, the easiest way to understand it is just by looking at the probability distribution function chart for coin tosses - you'll immediately see that mean has the highest chance of happenning, so exactly 100 is the most likely outcome
voakbasda 11mo ago
I would guess that a single trial of 200 flips can be treated as one event, so getting 100/100 is but one outcome. It may be the most likely individual outcome, but the odds of getting that exact result feel less likely than all of the other possible outcomes. The 100/100 case should be seen the most over repeated trials, but only marginally over other nearby results. Intuitively, this seems right to me, but sometime statistics do not follow intuition.
shakow 11mo ago
> every single other result is even less likely. But the summed probability of the “not too far away results” is much higher, i.e. P([93, 107]\{100}) > P([100]). So if you only shoot 100/100 with your coin, that's definitely weird.
- fainpul 11mo ago
  > So if you only shoot 100/100 with your coin, that's definitely weird. Not if you only try once.
  - kalaksi 11mo ago
    I'm sorry, but try what once? 200 flips once?
    - dredmorbius 11mo ago
      In statistics, various examples (e.g., coin flips) often stand in for other activities which might prove expensive or infeasible to make repeated tries of. For "coin flips", read: human lives, financial investments, scientific observations, historical observations (how many distinct historical analogues are available to you), dating (see, e.g., "the secretary problem" or similar optimal stopping / search bounding problems). With sufficiently low-numbered trial phenomena, statistics gets weird. A classic example would be the anthropic principle: how is it that the Universe is so perfectly suited for human beings, a life-form which can contemplate why the Universe it so perfectly suited for it? Well, if the Universe were not so suited ... we wouldn't be here to ponder that question. The US judge Richard Posner made a similar observation in his book "Catastrophe: Risk and Response" tackles the common objection to doomsday predictions that all have so far proved false. But then, of all the worlds in which a mass extinction event has wiped out all life prior to the emergence of a technologically-advanced species, there would be no (indegenous) witnesses to the fact. We are only here to ponder that question because utter annihilation did not occur. As Posner writes: By definition, all but the last doomsday prediction is false. Yet it does not follow, as many seem to think, that all doomsday predictions must be false; what follow is only that all such predictions but one are false. -Richard A. Posner, Catastrophe: Risk and Response, p. 13. <https://archive.org/details/catastropheriskr00posn/page/13/mode/2up https://archive.org/details/catastropheriskr00posn/page/13/m...>
dooglius 11mo ago
"fair coin" refers to both the probability of heads and tails being equal (which is still justified) as well as the trials being independent (unlikely with 100/200; more likely the "coin" is some imperfect PRNG in a loop)
- tshaddox 11mo ago
  > more likely the "coin" is some imperfect PRNG in a loop "More likely"? How can you even estimate the likelihood of the coin being "an imperfect PRNG" based on a single trial of 200 flips?
  - ajuc 11mo ago
    Bayesian vs frequentist in a nutshell :) If a student was tasked with determining some physical constant with an experiment and they got it exactly right to 20th decimal place - I'll check their data twice or thrice. Just saying. You continue believing it was the most likely value ;)
  - dooglius 11mo ago
    You can use combinatorics to calculate the likelihood. If your PRNG is in a cycle of length N in its state space (assuming N>200), and half the state space corresponds to heads (vs tails), then the likelihood would be (N/2 choose 100)^2/(N choose 200) versus your baseline likelihood (for a truly random coin) of (200 choose 100)/2^200. Graphing here https://www.wolframalpha.com/input?i=graph+%28%28N%2F2+choose+100%29%5E2%2F%28N+choose+200%29%29+and+%28%28200+choose+100%29%2F2%5E200%29+from+500+to+5000 https://www.wolframalpha.com/input?i=graph+%28%28N%2F2+choos... and it does look like it's only a slight improvement in likelihood, so I did overstate the claim. A more interesting case would be to look at some self-correcting physical process.
buildsjets 11mo ago
A truly fair coin would never land heads or tails. It would land standing on it's edge, every single time.
ptrl600 11mo ago
Low kolmogorov complexity equals suspicious

by fainpul 11mo ago

Joker_vD 11mo ago

Well, yes. But the expected deviation from the mean is still ≈7.07. And the probability that the outcome will be either 93/107 or 107/93 is (slightly) higher than the outcome being exactly 100.

fainpul 11mo ago
But those are 2 results. 100 / 100 is more likely than 93 / 107 (or any other specific result) is what I'm saying.
- Joker_vD 11mo ago
  Depends what you count as a result, I guess. "There is exactly N flips of a single kind" is also a viable definition, just as "The exact sequence of flips was x_0, x_1, ... x_199" is.
- paddleon 11mo ago
  You can tell the difference between 93 heads, 107 tails and 93 tails, 107 heads but not between 100 heads, 100 tails and 100 tails, 100 heads. So while those are two results (93/107 and 107/93), they really only count as two separate outcomes if you pre-specify that the first number is heads. If instead you consider symmetries, where there are 2 ways to get 93/107 and only one wall to get 100/100, then you have more likelihood for the 93/107 outcome because you have two ways to get it.
  - deleted 11mo ago
    [deleted]
- kalaksi 11mo ago
  I don't think that makes 100 / 100 the most likely result if you flip a coin 200 times. It's not about 100 / 100 vs. another single possible result. It's about 100 / 100 vs. NOT 100 / 100, which includes all other possible results other than 100 / 100.
  - andoando 11mo ago
    If you flip a coin 200 times, and repeat that a billion times, 100/100 will be more likely than any other ratio. It will be a bell curve with 100/100 as the peak
eviks 11mo ago
Why not go one abstraction further and go expected deviation from deviation? Probably the word "expected" plays a mind trick? "Expected" doesn't mean the probability increases, the easiest way to understand it is just by looking at the probability distribution function chart for coin tosses - you'll immediately see that mean has the highest chance of happenning, so exactly 100 is the most likely outcome

voakbasda 11mo ago

I would guess that a single trial of 200 flips can be treated as one event, so getting 100/100 is but one outcome. It may be the most likely individual outcome, but the odds of getting that exact result feel less likely than all of the other possible outcomes. The 100/100 case should be seen the most over repeated trials, but only marginally over other nearby results. Intuitively, this seems right to me, but sometime statistics do not follow intuition.

shakow 11mo ago

> every single other result is even less likely. But the summed probability of the “not too far away results” is much higher, i.e. P([93, 107]\{100}) > P([100]). So if you only shoot 100/100 with your coin, that's definitely weird.

fainpul 11mo ago
> So if you only shoot 100/100 with your coin, that's definitely weird. Not if you only try once.
- kalaksi 11mo ago
  I'm sorry, but try what once? 200 flips once?
  - dredmorbius 11mo ago
    In statistics, various examples (e.g., coin flips) often stand in for other activities which might prove expensive or infeasible to make repeated tries of. For "coin flips", read: human lives, financial investments, scientific observations, historical observations (how many distinct historical analogues are available to you), dating (see, e.g., "the secretary problem" or similar optimal stopping / search bounding problems). With sufficiently low-numbered trial phenomena, statistics gets weird. A classic example would be the anthropic principle: how is it that the Universe is so perfectly suited for human beings, a life-form which can contemplate why the Universe it so perfectly suited for it? Well, if the Universe were not so suited ... we wouldn't be here to ponder that question. The US judge Richard Posner made a similar observation in his book "Catastrophe: Risk and Response" tackles the common objection to doomsday predictions that all have so far proved false. But then, of all the worlds in which a mass extinction event has wiped out all life prior to the emergence of a technologically-advanced species, there would be no (indegenous) witnesses to the fact. We are only here to ponder that question because utter annihilation did not occur. As Posner writes: By definition, all but the last doomsday prediction is false. Yet it does not follow, as many seem to think, that all doomsday predictions must be false; what follow is only that all such predictions but one are false. -Richard A. Posner, Catastrophe: Risk and Response, p. 13. <https://archive.org/details/catastropheriskr00posn/page/13/mode/2up https://archive.org/details/catastropheriskr00posn/page/13/m...>

dooglius 11mo ago

"fair coin" refers to both the probability of heads and tails being equal (which is still justified) as well as the trials being independent (unlikely with 100/200; more likely the "coin" is some imperfect PRNG in a loop)

tshaddox 11mo ago
> more likely the "coin" is some imperfect PRNG in a loop "More likely"? How can you even estimate the likelihood of the coin being "an imperfect PRNG" based on a single trial of 200 flips?
- ajuc 11mo ago
  Bayesian vs frequentist in a nutshell :) If a student was tasked with determining some physical constant with an experiment and they got it exactly right to 20th decimal place - I'll check their data twice or thrice. Just saying. You continue believing it was the most likely value ;)
- dooglius 11mo ago
  You can use combinatorics to calculate the likelihood. If your PRNG is in a cycle of length N in its state space (assuming N>200), and half the state space corresponds to heads (vs tails), then the likelihood would be (N/2 choose 100)^2/(N choose 200) versus your baseline likelihood (for a truly random coin) of (200 choose 100)/2^200. Graphing here https://www.wolframalpha.com/input?i=graph+%28%28N%2F2+choose+100%29%5E2%2F%28N+choose+200%29%29+and+%28%28200+choose+100%29%2F2%5E200%29+from+500+to+5000 https://www.wolframalpha.com/input?i=graph+%28%28N%2F2+choos... and it does look like it's only a slight improvement in likelihood, so I did overstate the claim. A more interesting case would be to look at some self-correcting physical process.

buildsjets 11mo ago

A truly fair coin would never land heads or tails. It would land standing on it's edge, every single time.

ptrl600 11mo ago

Low kolmogorov complexity equals suspicious