# Probability; what is "the long run"?

On a roulette table with a single green zero the probability of the ball landing in a red pocket is 18/37 or 19/18 against (or approx 49% - with the odds for it landing in a non-red pocket (black or green) being approx 51%.

Probability theory tells us that although the ball will, at times, land in a non-red pocket several times in succesion, and, at times, many times in succession, in the LONG RUN it will land there approx 51% of the time (assuming an unbiased wheel etc).

But what is the LONG RUN?

A gambler can bet on red only to see the ball land in black (or green) say, 10 times in a row. Another gambler may see the ball land in black (or green) 15 or 20 times in a row (a freakish occurrence for some, but unremarkable for the mathematician – or experienced croupier).

Q. What is the ‘record’ for successive non-reds in actual play over the few centuries that roulette has been around?

Q. As a thought experiment, if we had monitored an unbiased wheel (with all other factors not causing any bias either) for the last two or three centuries what could the ‘record’ be in this case for successive non-reds?

Q. Does probability theory suggest that if we played for a long enough period of time we would see a hundred non-reds in succession? A thousand? Million? Billion, trillion etc?

Q. How can we predict when the LONG RUN (whatever that may be) will show us the true odds realised, ie, when we see there has been approximately 49% reds, 51% non-reds?

Q. If it’s the case that in theory we could see black come up say, a million times in a row (or more), is it true that (given sufficient spins) it would be the case in practice?

Dr. Courtney
Gold Member
2020 Award
The "long runs" means a great many trials. If you are trying to measure the occurances of an event with close to a 50% probability, you do not need as large a number of trials as when trying to measure the occurances of events with much smaller probabilities. The math is easier with coin flips.

Obtaining heads n times in a row has a probability of (1/2)^n. So the probability of 10 heads in a row is 1/1024.

• Cliff Hanley
phinds
Gold Member
Q. Does probability theory suggest that if we played for a long enough period of time we would see a hundred non-reds in succession? A thousand? Million? Billion, trillion etc?
?
Yes. Why would it not?

• Cliff Hanley
Stephen Tashi
Probability theory tells us that although the ball will, at times, land in a non-red pocket several times in succesion, and, at times, many times in succession, in the LONG RUN it will land there approx 51% of the time (assuming an unbiased wheel etc).

Technically, probability theory gives you no guarantees about any event (or series of events) actually happening. Probability theory merely uses the given probabilities to assign probabilities to other events and series of events.

When people assert that some event will happen in the long run, this is an assertion about the physics or other applied science involved in a problem, not a theorem of mathematical probability theory. The best mathematical probability theory can do in such situations is to say the limit of the probability of an event approaches 1 as the "length" of the "long run" approaches infinity.

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• Cliff Hanley
pbuk
Gold Member
Q. What is the ‘record’ for successive non-reds in actual play over the few centuries that roulette has been around?
You can google this, although I'm not sure how reliable the answers would be.

Q. As a thought experiment, if we had monitored an unbiased wheel (with all other factors not causing any bias either) for the last two or three centuries what could the ‘record’ be in this case for successive non-reds?
A wheel spun once a minute for 300 years will spin about 130 million times. The chance of 28 successive non-reds is about 1 in 127 million. However this doesn't mean that a run of 28 will happen, or that a run of more than 28 will not happen.

Q. Does probability theory suggest that if we played for a long enough period of time we would see a hundred non-reds in succession? A thousand? Million? Billion, trillion etc?
Yes: if everyone on Earth spent their whole lives playing roulette until the Earth's atmosphere is burned off by the Sun they are likely to see a hundred non-reds, but a thousand are unlikely before the universe reaches heat death (caution - I did these calculations rather carelessly).

Q. How can we predict when the LONG RUN (whatever that may be) will show us the true odds realised, ie, when we see there has been approximately 49% reds, 51% non-reds?
We can't, but we can say that the more trials we do the observed proportion is more likely to approximate the theoretical proportion more closely.

Q. If it’s the case that in theory we could see black come up say, a million times in a row (or more), is it true that (given sufficient spins) it would be the case in practice?
See the above comment on heat death.

You would gain more understanding by learning about this section of probability (binomial probability/Bernouilli trials) and doing the calculations yourself.

• Cliff Hanley and HallsofIvy
FactChecker
Gold Member
"The long run" depends on how close you want to get to 18/37. Even then, there is only a probability that it will get as close as you specify. So you have to frame the question this way: "How large of a sample size would it take so that the probability of the sample result being within xxx of its theoretical value is yyy?". The answer to that question would give you the sample size that you could call "the long run" for that case.

Suppose you want to say that there is a probability of 95% that it is within 0.01 of 18/37. Then there is an equation that tells you how many trials that would take. So it tells you what "the long run" would mean for that case.

• Cliff Hanley
WWGD
Gold Member
I think the long run here would be described by the LLN --Law of Large Numbers.

• Cliff Hanley
gill1109
Gold Member
Probability theory tells us that if we play infinitely often we will certainly get to see, *infinitely* many times, a hundred non-reds in succession. And a thousand. And a million. And a billion, and a trillion.

You name it, you will get it ... with probability 1, infinitely many times.

The strong law of large numbers.

• Cliff Hanley
FactChecker
Gold Member
I think the long run here would be described by the LLN --Law of Large Numbers.
That is just replacing the vague term 'long run" with the equally vague term "large numbers". So it begs the question "What is large?". In many cases like the roulette table of the OP, there are actual numbers that can be calculated if the question is asked correctly:
Given a confidence level, say 95%, and a desired accuracy, say 0.1, what is the sample size, N. that would give a sample accuracy of 0.1 with 95% confidence?

• Cliff Hanley
FactChecker
Gold Member
Probability theory tells us that if we play infinitely often we will certainly get to see, *infinitely* many times, a hundred non-reds in succession. And a thousand. And a million. And a billion, and a trillion.

You name it, you will get it ... with probability 1, infinitely many times.

The strong law of large numbers.
For answering the OP, I think it's important to add that this is not a contradiction of the Law of Large Numbers. The probability of an unusual sequence, say 1000 reds in succession, is so small that there are almost certainly a huge number of more normal results before that happens. So when the 1000 reds eventually occurs, it almost certainly does not effect the sample average very much.

• Cliff Hanley
phinds
Gold Member
Probability theory tells us that if we play infinitely often we will certainly get to see, *infinitely* many times, a hundred non-reds in succession. And a thousand. And a million. And a billion, and a trillion.
Actually "certainly" is a stretch isn't it. Yes, the probability of getting any string you can name approaches 1 as the number of trials approaches infinity, but since we can't actually do an infinite number of trials, we can't ever get an absolute certainty (probability = 1.0)

• Cliff Hanley
Stephen Tashi
Actually "certainly" is a stretch isn't it. Yes, the probability of getting any string you can name approaches 1 as the number of trials approaches infinity, but since we can't actually do an infinite number of trials, we can't ever get an absolute certainty (probability = 1.0)

There is a further distinction between "actually" and "certainly". If we "actually" took a sample from a normal distribution and the value was 1.23 then an event with probability 1 ( namely the event "the value of the sample will not be 1.23) failed to "actually" happen.

phinds
Gold Member
There is a further distinction between "actually" and "certainly". If we "actually" took a sample from a normal distribution and the value was 1.23 then an event with probability 1 ( namely the event "the value of the sample will not be 1.23) failed to "actually" happen.
I have no idea what you just said / what it means.

• Cliff Hanley
Stephen Tashi
I have no idea what you just said / what it means.

I'm making the distinction between the statement "Event E occurs" (Or "Event E will occur") versus the statement "Event E has probability 1".

For a normally distributed random variable X, let D be the event "X = 1.23". Let E be the event "X is not equal to 1.23". The event E has probability 1.

A similar statement holds true for any particular numerical value v of X. The probability that "X is not equal to v" is 1.

As another example, we have to distinguish between the truth of a statement A and the event "A is true with probability 1" when doing mathematical proofs.

For example, in logic we have the pattern of reasoning:
Given:
If A then B
A is true
----
Conclude B is true.

However it is not a valid form of logical argument to say:
Given:
If A then B
A is true with probability 1
----
Conclude:
B is true

• Cliff Hanley
phinds
Gold Member
This still makes no sense to me but I'll take your word for it.

• Cliff Hanley
FactChecker
Gold Member
Actually "certainly" is a stretch isn't it. Yes, the probability of getting any string you can name approaches 1 as the number of trials approaches infinity, but since we can't actually do an infinite number of trials, we can't ever get an absolute certainty (probability = 1.0)
I have to disagree. For any number, N, we can always continue long enough for N+1 occurrences. The probability of infinitely many occurrences is 1 because the probability of finite occurrences is 0.

• Cliff Hanley
WWGD
Gold Member
That is just replacing the vague term 'long run" with the equally vague term "large numbers". So it begs the question "What is large?". In many cases like the roulette table of the OP, there are actual numbers that can be calculated if the question is asked correctly:
Given a confidence level, say 95%, and a desired accuracy, say 0.1, what is the sample size, N. that would give a sample accuracy of 0.1 with 95% confidence?

Of course, this is the best we can do, but at least LLN gives you a theoretical backing, and, given a level of approximation wanted, then one can compute.

• Cliff Hanley
phinds
Gold Member
@Stephen Tashi I do have one question and perhaps the answer to it will enlighten me as to the rest of your comments.

How is it that the statement "A is true with probability 1" is anything other than simply an excessively redundant way of saying "A is true" ?

• Cliff Hanley
mfb
Mentor
As examples for the 49%/51% question:
After 1000 rolls, the chance to be within 1% of this result (so somewhere from 48/52 to 50/50) is roughly 50%.
After 10000 rolls, the chance to be within 1% is about 95%.
After 100,000 rolls, the chance to be within 1% is larger than 99.9999999%.

After 1 million rolls, the chance to be within 0.1% (between 48.9/51.1 and 49.1/50.9) is about 95% and the chance to be more than 1% away is completely negligible.
After 100 million rolls, the chance to be within 0.01% (between 48.99/51.01 and 49.01/50.99) is about 95%.

How is it that the statement "A is true with probability 1" is anything other than simply an excessively redundant way of saying "A is true" ?
Draw a random number from a uniform distribution over the real numbers in the interval [0,1]. "The number is not 0.5" has probability 1, but it is not certain. It is almost certain.

• Cliff Hanley and FactChecker
Stephen Tashi
@Stephen Tashi I do have one question and perhaps the answer to it will enlighten me as to the rest of your comments.

How is it that the statement "A is true with probability 1" is anything other than simply an excessively redundant way of saying "A is true" ?

As mfb's example illustrates, the mathematical definition of probability is very technical. The mathematical definition of probability describes a situation where events are assigned numbers called "probabilities". It doesn't specify anything about whether events actually happen and it doesn't provide any guarantees about the frequency with which they happen.

A good way to understand the situation in an intuitive and philosophical manner is to consider the general problem of formulating a theory about "uncertainty". What we desire from theories for them to make definite statements and predictions. We don't want the theory itself to be "uncertain". So how can you say something "certain" about "uncertainty"?

Probability models uncertainty by assigning numbers to events. The assignment is something definite - e.g. "The probability that a fair coin lands heads is 1/2". The conclusions of the theory are definite - e.g. The probability of two heads in two independent tosses of the fair coin is 1/4". We can make definite statements because we are talking about "the probability of" an event instead of asserting something about the event happening without the clause "the probability of" attached to it. So the general pattern of results in probability theory is: "If the probability of ... is such-and-such then the probability of ... is so-and-so".

Because the conclusions of probability speak of the "probability of" events, the conclusions of probability theory are not conclusions about the events without the modifying phrase "probability of" attached to the event.

People who apply probability theory to practical problems may assert that "E has probability 1" amounts to the same thing as "E happens" and this is a valid claim in many practical situations. However this claim is not a consequence of the mathematical theory of probability. The claim must be supported by some additional facts or assumptions about the practical situation being considered.

• jim mcnamara and Cliff Hanley
How is it that the statement "A is true with probability 1" is anything other than simply an excessively redundant way of saying "A is true" ?

It is a very different statement, and one whose difference is not usually taught well to students.
Let me change the wording to "A has probability 0" and "A is false". It's the same thing, but it's easier to explain it this way.
Here is the critical example. Pick any number between ##0## and ##1## at random (assuming every number has the same possibility = uniform). What is the probability you picked ##1/2##. Math says the probability is ##0##. But it's not impossible you picked ##1/2##. In fact, every number has probability ##0## of being picked. But you have to pick some number.

This is why something with probability ##0## is said in mathematics to be true "almost always" or "almost surely". The "almost" is very important!

• Cliff Hanley and FactChecker
Stephen Tashi
Here is the critical example. Pick any number between ##0## and ##1## at random (assuming every number has the same possibility = uniform).

It's worth pointing out that the formal theory of probability does not assert that we can take random samples. It only asserts that given a distribution, we can determine the probability of certain events.

Of course, in applying probability theory we deal with situations where random samples are actually taken. However, in such a situation if we consider statements of the form "If we take a random sample of .... then...(some disagreeable conclusion)" the disagreeable conclusion might occur because the premise "we take a random sample" is false. For example, in a given practical situation, one can debate whether it is possible to take a random sample from a uniform distribution on [0,1]. For example, if the sample is a reading from digital display on a voltmeter then it has limited precision.

• Cliff Hanley
phinds
Gold Member
It is a very different statement, and one whose difference is not usually taught well to students.
Let me change the wording to "A has probability 0" and "A is false". It's the same thing, but it's easier to explain it this way.
Here is the critical example. Pick any number between ##0## and ##1## at random (assuming every number has the same possibility = uniform). What is the probability you picked ##1/2##. Math says the probability is ##0##. But it's not impossible you picked ##1/2##. In fact, every number has probability ##0## of being picked. But you have to pick some number.

This is why something with probability ##0## is said in mathematics to be true "almost always" or "almost surely". The "almost" is very important!
OK, I get it, assuming that in the last sentence, you have a typo and meant probability 1 (or, alternatively, "almost always false"), yes?

• Cliff Hanley
gill1109
Gold Member
Actually "certainly" is a stretch isn't it. Yes, the probability of getting any string you can name approaches 1 as the number of trials approaches infinity, but since we can't actually do an infinite number of trials, we can't ever get an absolute certainty (probability = 1.0)
You can make a mathematical model for an infinite number of trials and in that mathematical model there is probability 1 that any particular string will be repeated infinitely many times. As a consequence of the strong law of large numbers. To be sure, the mathematical result is obtained by showing that the probability any particular string is repeated at least some particular number of times in N trials converges to 1 as N tends to infinity.

phinds