Probability Conundrum
Say in Keno you draw 1, 3, 9, 5, 6, 15, 30, 33. By itself the probability of drawing these eight numbers is 1 in one billion. Anyone who receive this set of numbers is going to say to receive this set of numbers is so improbable cheating must have occurred!
Yet if you realize for any draw you are bound to receive one set of of this numbers. So for Any result you are going to say cheating Must have occurred since the chance of getting this set of numbers is one in a billion.
How you resolve this conundrum
I'm not sure why you ask about this set of numbers in particular. Maybe that they are all 33 or less. The chances of that are 1 in 2088.
I hope this helps, but I'm not not sure what the conundrum is.
Quote: niuniukingDoes anyone want to solve this?
Say in Keno you draw 1, 3, 9, 5, 6, 15, 30, 33. By itself the probability of drawing these eight numbers is 1 in one billion. Anyone who receive this set of numbers is going to say to receive this set of numbers is so improbable cheating must have occurred!
Yet if you realize for any draw you are bound to receive one set of of this numbers. So for Any result you are going to say cheating Must have occurred since the chance of getting this set of numbers is one in a billion.
How you resolve this conundrum
If I understand what you're getting at, the flaw is in your thinking that an extremely low probability of something happening is evidence of cheating. That's simply not the case. As you state, if you're going to draw 8 numbers from 80, it has to be one combination. So determining whether cheating is happening should be done on the basis of whether the draw mechanism was fair, not if the outcome was fair.
In Another thread I posted a question about how 7 different casinos each independently had a 1 in 500 odds outcome opened. Forumers replied cheating Must have occurred since the odds of casinos independently having 1 out of 500 each is 1/500^7, more than 1 in one Trillion!
Yet if you think carefully about it, EACH casino will deliver one result yes? say A roulette outcome of 36 twice. The other casino an outcome of 15 twice etc. So practically to ANY result you are then going to accuse this result to having occurred as a result of cheating.
Counter this allegation
https://wizardofvegas.com/forum/questions-and-answers/math/31513-niu-niu/13/#post791582
Quote: rsactuaryIf I understand what you're getting at, the flaw is in your thinking that an extremely low probability of something happening is evidence of cheating. That's simply not the case. As you state, if you're going to draw 8 numbers from 80, it has to be one combination. So determining whether cheating is happening should be done on the basis of whether the draw mechanism was fair, not if the outcome was fair.
Quote: niuniukingDoes anyone want to solve this?
Say in Keno you draw 1, 3, 9, 5, 6, 15, 30, 33. By itself the probability of drawing these eight numbers is 1 in one billion. Anyone who receive this set of numbers is going to say to receive this set of numbers is so improbable cheating must have occurred!
Yet if you realize for any draw you are bound to receive one set of of this numbers. So for Any result you are going to say cheating Must have occurred since the chance of getting this set of numbers is one in a billion.
How you resolve this conundrum
The chance of drawing eight numbers is 100%.
When looking at past numbers drawn, do they pass tests for randomness? When seeing new numbers drawn, is there any predictable patters to them? The answers to those questions resolve the conundrum.
The chance of getting this particular set is 1 in 1 trillion. Also, say you get only 1 draw so you cannot compare. A person may reason to get this particular set is extremely improbable. Which is why this conundrum exist. Yet we must have a particular result.
Quote: niuniukingThe conundrum is this. Someone who receives this or any result is going to say" Hey it's so improbable to receive this numbers. There must have cheating going on for you to receive this numbers against the odds!
In Another thread I posted a question about how 7 different casinos each independently had a 1 in 500 odds outcome opened. Forumers replied cheating Must have occurred since the odds of casinos independently having 1 out of 500 each is 1/500^7, more than 1 in one Trillion!
Yet if you think carefully about it, EACH casino will deliver one result yes? say A roulette outcome of 36 twice. The other casino an outcome of 15 twice etc. So practically to ANY result you are then going to accuse this result to having occurred as a result of cheating.
Counter this allegation
You don't specify in the "seven casinos" version under what conditions the 1 in 500 events happened.
If it is a case of "each one picked a number from 1 to 500," then yes, each had a 1 in 500 event occur, but that's an expected result in this case as every possible outcome is seven separate 1 in 500 results.
On the other hand, if it is seven specific events stated in advance - say, "If the number 123 is chosen, all player bets lose," and all seven casinos just happened to have their next number be 123, then there is legitimate reason to suspect the randomness of the selections.
Quote: niuniukingYou missed the point.
The chance of getting this particular set is 1 in 1 trillion. Also, say you get only 1 draw so you cannot compare. A person may reason to get this particular set is extremely improbable. Which is why this conundrum exist. Yet we must have a particular result.
It seems like your point is that the probability of an improbable event is 100%. If you find that to be an unresolvable conundrum, that is on you. The rest of us have accepted it.
Quote: niuniukingThere is still the conundrum isn't it? The 1 in 500 result. What's the probability that there is this specific one in a trillion result? Yet there Must be a result and Any result is one in a trillion
When there are one trillion, one-in-a-trillion shots, we can look at it like this 0.000000000001 x 1,000,000,000,000. That equals a 100% chance of getting a result. Therefore, getting a result, is not be seen as a conundrum by most of us.
If there is only one, one-in-a-trillion shots, and it happens to hit, then there is something interesting.
Quote: niuniukingI do not have this point. Anyways for the thread in question. a 1/500 event occurred INDEPENDENTLY at each of seven Different casino. Am I to suspect cheating is going on? Yes or no? And this meant us losing money.
Based only on this information, I would not suspect cheating. If there was more relevant information available, it might be able to prove cheating.
You can see if you get many high counts it is just a result, an unlikely one. Over here experts regard this as a red flag. Somehow then in this thread it is not. When is it when it isnt?
Quote: niuniukingYou missed the point.
The chance of getting this particular set is 1 in 1 trillion. Also, say you get only 1 draw so you cannot compare. A person may reason to get this particular set is extremely improbable. Which is why this conundrum exist. Yet we must have a particular result.
You already said it: We must have a particular result.
I don't understand the focus on electronic games for this question as this is true of anything. I'll give you one example: Suppose you are at a Craps Table and the dice are rolled twenty times, what is the probability that all twenty rolls are sevens?
(1/6)^20 = 2.7351112e-16 or 1 in 3,656,158,500,000,000
So, we would say, "That's ridiculous. That's absolutely never going to happen."
However, it has to be accepted that seven is the single most likely result of a roll of two dice; we know this to be true. Because seven is the most likely single roll, if we are looking at the outcomes as being in a specific order, twenty sevens in a row is the most likely individual sequential outcome possible. For example:
7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7
Is a more likely specific sequence than:
7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7
Because a seven is more likely than a four for that individual roll.
But, that's not a conundrum of any sort, it's just the fundamental function of probability. One of the things some people have a tough time getting their heads around is the fact that we notice certain things just because the particular series of outcomes stands out in a purely subjective way. Objectively speaking, a particular series is typically no less likely than some other series would have been with the same number of individual results in the sequence.
Again, isolating twenty rolls of two dice, we would think twenty sevens is ridiculous...but if we are looking at the twenty outcomes in sequential terms, it's technically the most likely individual thing of all possible things that can happen.
Quote: niuniukingHow to differentiate "just a result" and cheating redflag can you clearly tell me?
No one will be able to ‘clearly tell you’. You call me up on the phone. Ask me to turn over 8 cards and read them out loud.
2C. 8H. KS. KC. 5D. 9H. 10S. QC. Are you going to say ‘no way’! The odds are around 4 trillion to 1 against me having drawn that specific sequence. But you believing whether I drew that is more dependent on what you think of me! Do you think I had a motive to lie? If you did, then you will disbelieve me. If you think I didn’t, then you will just believe me.
I translate that to your casino question. If it is a heavily regulated ‘brick and mortar’ casino, a single 1 in 500 chance does not raise my eyebrows. If 7 have the same result simultaneously, I believe it is because of chicanery. If a single 1 in 500 chance occurs at an on line casino, I believe it is more likely chicanery than honest.
So context is important in deciding if the event is more likely to have happened or more likely to be a lie, or a scam.
I don't know if my credentials as a retired professor of mathematics with a specialty in probability and statistics makes a difference to you, but it is a very good question and one that I answered many times as a professor.
Your very existence is far more unlikely than any Keno or lottery numbers coming up, just think of the chain of events that led to you. And yet here you are. Given that you exist, what is the probability you exist? Reframing this as a conditional probability is the only way to make sense of the question and its obvious answer.
Quote: niuniukinghttps://www.blackjackinfo.com/community/threads/the-casino-is-cheating-confront-or-exit-quietly.21941/page-2
You can see if you get many high counts it is just a result, an unlikely one. Over here experts regard this as a red flag. Somehow then in this thread it is not. When is it when it isnt?
I already gave an answer, when I asked what the repeatability and predictability of the upcoming events would be. If we see a few shoes start by going into a high positive count, we might predict the upcoming shoes also start by going into a high count. Then, if it does happen, (predictable and repeatable) we would have evidence to speculate cheating. That is definitely not the only method to detect cheating, but it is a straightforward one. It looks like they even provided even more evidence to support cheating, by saying the casino did not follow typical protocols with the cards. When you do not provide that level of information, and instead just say "a 1 in sextillion event happened at seven different casinos", there is no red flag.
When I brought this up before, you immediately dismissed it as not getting the point. At what point do you stop turning everything into an argument? That's the conundrum I see here.
Quote: SOOPOONo one will be able to ‘clearly tell you’. You call me up on the phone. Ask me to turn over 8 cards and read them out loud.
2C. 8H. KS. KC. 5D. 9H. 10S. QC. Are you going to say ‘no way’! The odds are around 4 trillion to 1 against me having drawn that specific sequence. But you believing whether I drew that is more dependent on what you think of me! Do you think I had a motive to lie? If you did, then you will disbelieve me. If you think I didn’t, then you will just believe me.
I translate that to your casino question. If it is a heavily regulated ‘brick and mortar’ casino, a single 1 in 500 chance does not raise my eyebrows. If 7 have the same result simultaneously, I believe it is because of chicanery. If a single 1 in 500 chance occurs at an on line casino, I believe it is more likely chicanery than honest.
So context is important in deciding if the event is more likely to have happened or more likely to be a lie, or a scam.
Depends on the 1 in 500 chance in question! I maintain that some results would lead you to conclude that the casino is honest!
Damn. I was gonna say that.Quote: Mission146Put another way, "$#!t happens,"
In fact, I did — on dozens of other similar threads in the last ~11 years. 🤪
Quote: niuniukingThere is still the conundrum isn't it? The 1 in 500 result. What's the probability that there is this specific one in a trillion result? Yet there Must be a result and Any result is one in a trillion
It's not a conundrum at all. While the probability of a particular "one in a trillion" result occuring is one in a trillion, the probability that any of one trillion results of a given event, each of which is distinct and has a probability of one in a trillion, is one.
The "conumdrum" reminds me of this Dilbert comic strip:
This is beginning to sound like Zeno's "Achilles and the Tortoise" paradox, where, if a tortoise is given a head start in a race against Achilles, then Achilles can never reach the tortoise, as by the time Achilles gets to the point where the tortoise started, the tortoise will have moved forward, and by the time Achilles reaches the tortoise's point, the tortoise will have moved forward some more.
You made an earlier reference to a thread where you posted something about seven casinos each having a 1 in 500 event happen. Could you provide a link to the post, so I can apply it in context to this supposed conundrum?
Quote: teliotThe way I used to address this when I taught at the University was to say that probabilities are only meaningful in advance of an event, or prior to an event. After the fact, the probability that the thing that happened is the thing that happened is equal to 1. In other words, after the fact it becomes a conditional probability. Given that the thing happened, what is the probability that it happened. The answer is 1.
I don't know if my credentials as a retired professor of mathematics with a specialty in probability and statistics makes a difference to you, but it is a very good question and one that I answered many times as a professor.
Your very existence is far more unlikely than any Keno or lottery numbers coming up, just think of the chain of events that led to you. And yet here you are. Given that you exist, what is the probability you exist? Reframing this as a conditional probability is the only way to make sense of the question and its obvious answer.
Quote: ThatDonGuyIt's not a conundrum
You made an earlier reference to a thread where you posted something about seven casinos each having a 1 in 500 event happen. Could you provide a link to the post, so I can apply it in context to this supposed conundrum?
But if the original poster had talked about testing for fairness, I would have answered the question differently. For example a chi-square test on number distribution could be designed in advance of looking at the data, it would be the same test no matter which data you were handed, and the particular data may pass or fail that test. But I cannot design a test for a particular bias I already see in the data that is unique to that data. In that case I would need future data.
It's a similar concept, hypothesis testing requires unbiased test design.
One thing, when I ran certified Fair gambling, I never disclosed to my clients which statistical tests I would conduct, so they could not possibly know how they did they sent me would be examined. This makes cheating much harder.
Before I went to Niu casino or others to BJ we did Not expect such results. So in our case should we suspect cheating given we did not expect them.
Quote: teliotThat was not the question the original poster made. But in general hypothesis testing asks about future events. So for example, before looking at data from an online casino, you can make the hypothesis that the casino is fair and design tests to disprove the null hypothesis. But what you can't do is design specific tests based on the data you are handed.
But if the original poster had talked about testing for fairness, I would have answered the question differently. For example a chi-square test on number distribution could be designed in advance of looking at the data, it would be the same test no matter which data you were handed, and the particular data may pass or fail that test. But I cannot design a test for a particular bias I already see in the data that is unique to that data. In that case I would need future data.
It's a similar concept, hypothesis testing requires unbiased test design.
One thing, when I ran certified Fair gambling, I never disclosed to my clients which statistical tests I would conduct, so they could not possibly know how they did they sent me would be examined. This makes cheating much harder.
As for the previous post, "cheating" is too broad. If you said, "I did not expect the dealer to get 30 blackjacks in the first 100 hands" and then you played and the dealer got 30 blackjacks in the 100 hands, then I would suspect cheating. I would then conduct a test, "hand outcome test" (which was part of my standard suite) on a fresh set of data and chi-test on hand outcome distribution for the dealer. You can use past experience to suspect cheating, but the tests you use should not be extraordinary and should be forward looking. Of course, if you have a huge amount of data, you can develop a belief in a particular bias based on examining a small subset of the data, then test it on the full set of data. But you should never use ALL the data to be the basis for which test to design.
Let me give you example of what can go wrong. Many years ago I had a client who was sure a casino was cheating. I did a "starting hand total" chi-squared test for the dealer. In that, we found that the result indicated that game was fair and passed that test. But, in particular he noted that the dealer had a slightly above average number of 10, 11, 19 and 20 as starting hands over the hands he played. He then requested a "10/11/19/20" starting hand for the dealer test" on his data which I unfortunately conducted. That is, you simply test if the starting hand is one of those four hands vs. any other hand (there's a bit more to it than this, but it doesn't matter). Of course, the output of the test was a huge fail for the casino. I had a long back and forth with the casino, who pointed out the error in test design and ultimately I had to agree with them, much to the chagrin of my client. I returned my client's fee, and then he accused me of working for a cheating casino. Of course, if you lose, then you probably lost because you happened to fall on the bad side of a run of good dealer hands, so the test was nonsense.
That's what can go wrong.
Over the years I did some forensic auditing as well on a number of online casinos. In one instance, I was able to prove a 50/50 outcome was 48/52 and even demonstrate how it was likely coded. In another, I tracked down that the Craps software had a glitch that allowed the player to remove his Pass Line "points" without penalty (this gave a 11.1% player edge). So, in an informal sense, you can look at data and draw conclusions, but you don't want to be an expert witness testifying to such results.
So now the results After I went in with this Inquisitive open mind. turns out to be 1/500 for four of them and fifth reaching it. According to Eliot then I Should suspect t cheating? After all it is not a conditional probability, where, say, one day you chanced upon a casino and realised an already opened result is 1/500
Quote: teliotI was answering the OP. I knew even when I answered it that "cheating" was going to be brought up. Too bad. It's such a nice philosophical question on its own.
As for the previous post, "cheating" is too broad. If you said, "I did not expect the dealer to get 30 blackjacks in the first 100 hands" and then you played and the dealer got 30 blackjacks in the 100 hands, then I would suspect cheating. I would then conduct a test, "hand outcome test" (which was part of my standard suite) on a fresh set of data and chi-test on hand outcome distribution for the dealer. You can use past experience to suspect cheating, but the tests you use should not be extraordinary and should be forward looking. Of course, if you have a huge amount of data, you can develop a belief in a particular bias based on examining a small subset of the data, then test it on the full set of data. But you should never use ALL the data to be the basis for which test to design.
Let me give you example of what can go wrong. Many years ago I had a client who was sure a casino was cheating. I did a "starting hand total" chi-squared test for the dealer. In that, we found that the result indicated that game was fair and passed that test. But, in particular he noted that the dealer had a slightly above average number of 10, 11, 19 and 20 as starting hands over the hands he played. He then requested a "10/11/19/20" starting hand for the dealer test" on his data which I unfortunately conducted. That is, you simply test if the starting hand is one of those four hands vs. any other hand (there's a bit more to it than this, but it doesn't matter). Of course, the output of the test was a huge fail for the casino. I had a long back and forth with the casino, who pointed out the error in test design and ultimately I had to agree with them, much to the chagrin of my client. I returned my client's fee, and then he accused me of working for a cheating casino. Of course, if you lose, then you probably lost because you happened to fall on the bad side of a run of good dealer hands, so the test was nonsense.
That's what can go wrong.
Over the years I did some forensic auditing as well on a number of online casinos. In one instance, I was able to prove a 50/50 outcome was 48/52 and even demonstrate how it was likely coded. In another, I tracked down that the Craps software had a glitch that allowed the player to remove his Pass Line "points" without penalty (this gave a 11.1% player edge). So, in an informal sense, you can look at data and draw conclusions, but you don't want to be an expert witness testifying to such results.
Quote: tomchina123it is a long time i reopen this thread.
i come to say goodbye to the game of niuniu. because i think i was cheated.
i played in a big and famous casino in sihanouk. flat bet. 1:1, not house edge. no magnet in the dice as tested. i played 150 times, lost 40 bets.
after this, i felt very bad and made a simu. it is like once in more than 3 years. which means, if i make 150 bets every day, it happens only once in 3 years. why did it happen in that day?
cheating is very often in this city. but on that day, on that casino, i made the decision to say goodbye to niuniu. and very likely most dice-related games.
readers may think i am some one who is easy to be cheated. i should inform: in this city,i made good money last year ok money this year.
There are over 3 billion females in the world and every single one of those 3 billion women is not having sex with you tonight. Isn't that massively improbable? That all 3 billion women are doing the same thing?
None dare call it . . . conspiracy.
Quote: gordonm888It's like this with large numbers:
There are over 3 billion females in the world and every single one of those 3 billion women is not having sex with you tonight. Isn't that massively improbable? That all 3 billion women are doing the same thing?
None dare call it . . . conspiracy.
But if only 2,999,999,997 aren’t things are good!
The post above mentioning our very existence is less likely than any of these rare events. I do think that I met my first wife as a result of a lottery that placed students in dorm rooms. Had I been on a different floor or even a different dorm I wouldn’t have met the original SOOPOO. Thus no Son of SOOPOO. And I never would have lived in Buffalo. Etc.....
Quote: gordonm888There are over 3 billion females in the world and every single one of those 3 billion women is not having sex with you tonight. Isn't that massively improbable? That all 3 billion women are doing the same thing?
Not in my case. I would have scared off all 3 billion with trivia questions, carefully chosen to make them feel stupid. Well, I'd like to think I would have had a chance with Emma "Jeopardy" Boettcher.
On a related note, a normal sperm count per healthy ejaculation is about 40 to 80 million. Those are the odds that you were conceived. I don't sit there contemplating that as it had to be somebody. Why not me?
Mike, that is only one generation of probability. You are the result of countless generations of reproduction going back millions of years.Quote: WizardNot in my case. I would have scared off all 3 billion with trivia questions, carefully chosen to make them feel stupid. Well, I'd like to think I would have had a chance with Emma "Jeopardy" Boettcher.
On a related note, a normal sperm count per healthy ejaculation is about 40 to 80 million. Those are the odds that you were conceived. I don't sit there contemplating that as it had to be somebody. Why not me?
Quote: teliotMike, that is only one generation of probability. You are the result of countless generations of reproduction going back millions of years.
You're absolutely right.
