How should we evaluate improbable outcomes in a probabilistic system?

Question

Suppose I observe a highly improbable outcome while playing roulette - for example, 50 black results in a row, with a probability of 1/2^50. A mathematician would likely say the probability remains 1/2 for the next spin, since each event is independent.

From a physical or empirical perspective, however, if we see an outcome that is highly unlikely according to our mathematical model (like 50 black results in a row), it might be reasonable to suspect that there is something wrong with our model or with the device that is generating the outcomes (in this case, the roulette wheel). The physicist might argue that the device is biased in some way, or that some other physical factor (like a magnet) is influencing the results. And he might argue that the chance of black in next spin is 100%.

Now there a third way to answer this - if the wheel is broken, there is a chance it's not broken to give us blacks all the time. Maybe it's programmed to give white now, right when we are asked? In this case the correct answer to this - "Impossible to evaluate probability since we are unaware of the nature of the wheel's faulty behaviour (if we suppose there is one)?

As a philosopher seeking truth through reasoned inquiry, how should I evaluate this scenario? Should I trust the mathematically calculated 1/2 odds, or does the extremely improbable empirical result suggest the wheel is biased? Is it possible to integrate these perspectives?

More broadly, how do we address the tension between conceptual probability models and observations of real-world randomness that seem to defy the odds?

The probability of getting 1111111111 is exactly the same as getting 1001101001. You just overestimate the first one and underestimate the second. — RodolfoAP, Jul 24 '23 at 10:18
I am talking about probability of 50 blacks in a row, it has a probability of around 1 time in 12 trillion.. If there are other very low probabilities of other patterns, it doesn't change my question in the slightest. You are missing the point of the question. — , Jul 24 '23 at 10:23
Also when we're talking about the likelihood of getting a sequence with at least one zero compared to a sequence with all blacks, we're comparing two different sets (or "multitudes") of sequences.
The first set contains all possible sequences of 50 spins that result in all black. There's only one such sequence: "black, black, ..., black" (50 times). The second set contains all possible sequences of 50 spins that include at least one zero. This set is much larger since it includes any sequence with one or more zeroes and the rest black. BUT this not even what question is about — , Jul 24 '23 at 10:29
See: 'How improbable does an event have to be before we can say it didn't happen by chance?' https://philosophy.stackexchange.com/questions/94079/how-improbable-does-an-event-have-to-be-before-we-can-say-it-didnt-happen-by-ch/94082#94082 — CriglCragl, Jul 24 '23 at 16:14
Thank you. I know that in my case it's practically impossible since the chance is in the same category as Universe age in seconds — , Jul 24 '23 at 16:22
In practice, if you suspect that a device is giving results that are so anomalous that your model of it is wrong, then the best option is to examine it and find out exactly how it works. If a roulette table is biased (which they always are to some extent, because they are never perfectly level) or if a device gives results that depend on what measurements you make of it, you need to study its mechanism to find out why. Until you do, the probability of an outcome is unknown or has a wide range of possible values. — Bumble, Jul 24 '23 at 19:09
The example is apropos. What Age is this? Glad, it ain't the Stone Age. — Agent Smith, Jul 26 '23 at 19:03

score 1 · Answer 1 · answered Jul 26 '23 at 18:33

Probability is based on the history of prior events. Brand new roulette wheels have no history so the odds are based on the results of other identically constructed wheels. Each spin event creates an outcome that is added its history. As a result, probabilities are not static. They change each time the wheel is spun. The static nature of probabilities comes from the assumption that the static probability is from a number of events that approaches infinity. A cluster of 50 out of a near infinite event space seems insignificant. If the roulette wheel is brand new and, starting with the first roll, there are 50 blacks in a row, Then the actual calculated probability of a black occurring on the next roll is 100%. If it's just a cluster, then a much larger event history will be required to put it in context. So knowledge of an appropriate number of prior events influences any single analysis of an improbable event.

score 0 · Answer 2 · answered Jul 24 '23 at 10:34

First of all the odds are even worse as roulette tables have green 0 (or even 00) fields where you lose with both red and black.

Anyway in real life you'd simply define an "arbitrary" threshold where you'd still believe a deviation from the expectation, while if the value discrepancy is bigger than that you'd reject it.

And then you can compute your confusion matrix, you know rejection despite being true (false negative) or acceptance despite being false (false positive).

That doesn't mean that such a sequence is impossible or that the machine is not biased against you, it just means you set conditions for yourself in terms of what risk of being wrong you'd value more or less.

I know there are zeroes, I even didn't mention it to simplify the matter, but ty for reminding — , Jul 24 '23 at 10:35

score 0 · Answer 3 · answered Jul 24 '23 at 15:36

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The probability of 50 blacks in a row is exactly the same as that of any other sequence of 50 results. The roulette wheel is a chaotic device that cannot be biased. The ball is launched manually and it's completely impossible to launch it with sufficient precision to make any bias.

Darts is a game of skill, where novice players get random results like in roulette and experienced players get less random higher points biased results.

answered Jul 24 '23 at 15:36

Pertti Ruismäki

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It's a mere question of engineering to build a machine where ball would land in blacks. 2. As for another random sequence - It doesn't change my question whatsoever, I compare it not with a specific set of red and blacks, but with the whole multitude of outcomes that include BOTH black and red results.

Jul 24 '23 at 15:52

@SergZ. 1. A roulette wheel is engineered to produce random unbiased results. The zeros are there to ensure that the house always wins in the long run. 2. You cannot compare the probability of one result versus multiple other results. The result of 50 blacks is just as probable as 50 reds or any combination of reds and blacks. There is nothing special about 50 blacks in a row. It is just one of the equally probable sequences. – Pertti Ruismäki Jul 25 '23 at 05:29

score 0 · Answer 4 · 2023-07-24T19:42:48.347

The probability remains 1/2 only if you assume that the roulette wheel is not rigged. That probability, given that assumption, never changes.

What you’re really trying to get at is what the probability of the next result being black is without that assumption. Now, given that rigged roulette wheels are possible, it seems reasonable to conclude or atleast bet that the next result will still be black, and not at a 50% chance, but much higher. Why? Because the likelihood of getting 50 black results in a row given no cheating pales in comparison to the likelihood of getting them in a row given cheating. The latter likelihood, if rigged to always return blacks, would literally be 1.

Now, the exact “probability” of the next result being black remains undefined. Because of this, the answer to a follow up question being “how many black results in a row would you need before suspecting cheating?” remains undefined. It arguably does not have a correct answer. In the real world, the next result will be black or not. There is no “probability” in this binary outcome so attaching exact numbers to the final result is subjective.

However, it remains reasonable to conclude that there was cheating involved after 50 or 100 or 150 black results. This is because intuitively, given that we apriori know that rigging is possible and has happened atleast some number of times in the history of the world, the prior probability of rigged wheels seems to be higher than the probability of getting, say, 100 black results by chance. That is what makes the difference.

Note that if there was no apriori mechanism of rigging known, this conclusion would not be valid. For example, replace black results with coins, and imagine a person claims to be a psychic and control coin tosses with his mind. Then, even after a 100 straight heads, it would not be reasonable to conclude that he’s a psychic. This is because, apriori, one needs independent evidence of the direct sort to show that psychism is possible in the first place. Without this, one cannot say that the prior probability of “psychism” is higher than the probability of getting 100 heads in a row by chance, even if the latter is extremely improbable.

So to summarize, decisions should be made based on the relative difference in probabilities of certain hypotheses. It is only rational to conclude an alternative hypothesis if the prior probability of that hypothesis, in atleast some intuitive sense, is higher than the probability of that result by chance.

I can also add that we didn't see how this roulette behaved BEFORE our 50 spins. ;) — , Jul 24 '23 at 20:03
If you didn’t see how it behaved, that gives you more reason to think it was rigged. Since a presumably rigged roulette wheel is more likely to stay rigged if it is than to be rigged for a certain specific sequence of 50 spins. But again, part of this is just postulation, and it depends upon the exact mechanism and engineering done to rig a wheel. — , Jul 24 '23 at 20:16

score 0 · Answer 5 · answered Dec 29 '23 at 06:37

You should read up on The Null Hypothesis. H0, the null hypothesis, is that E, an event,(e.g. rolling 50 blacks), is just chance. H0 implies a certain limit to the frequency of the occurence of E (say < 1/1,000,000) by chance. If E occurs then at a frequency higher e.g. 10/1,000,000 and this is statistically significant (currently defined as P(10/1,000,000 is chance) <= 5%), we reject H0 i.e. E did not occur by chance and we're 95% confident about that.

How should we evaluate improbable outcomes in a probabilistic system?

5 Answers5