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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.