Author Topic: A Common Error in Probability  (Read 34797 times)

Bayes

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A Common Error in Probability
« on: April 25, 2015, 11:22:50 AM »
In my years of reading on forums, this is the one error I see time and time again. It concerns confusing the probability of a series with that of a single.

For example, there is a famous example in probability called the "Birthday Problem" which states that in a room of 23 people, there is a 50% chance that at least two will share a birthday. Does this suggest a system for roulette? what is the probability that in a sequence of spins, you will get at least one repeat?

If you do the math, it turns out that in any 8 spin sequence, there is a roughly 56% chance that there will be at least one repeat in the sequence. No problem with that, but the error occurs when you make a statement like this:

"So it appears that after 7 consecutive numbers have appeared and none are repeats there is a 56% chance that one of the 7  will repeat on the next spin."

This is saying that there is a 56% chance that you will get a win when betting on 7 numbers! but any casino which offered the equivalent of such odds would soon be out of business. The mistake lies in assuming that you can apply the probability of the series (7 spins with at least one repeat) to that of a single outcome. But once the 7 spins have gone, probability applies to the next spin only, so the original probability is now meaningless. All you can do is bet the last 7 numbers and hope that one of the last 7 repeats. What is the chance of that?

The answer is 7/37, no more and no less. If you doubt this, assume that the last 7 numbers were 17,1,32,25,8,12,28. The chance that 17 will hit on the next spin (and so result in a repeat) is 1/37, the chance that 1 will repeat is again 1/37. Similarly for each of the others. Since these are mutually exclusive outcomes, we can add the results, which gives 7/37.

You can indeed make a system out of the knowledge that there is at least one repeat with probability 56% in the last 8 spins, but in order for the probability to remain valid, you have to place your bets from spin 1, not after spin 7. So on spin 1 you put one chip on the last outcome, on spin 2 add another chip to whatever just hit, and so on, until you get a repeat (a win). But in that case, your profit will vary according to when you get the repeat, assuming you do get it. 56 times out of 100 you will indeed get at least one, but what you cannot say is that you will win 29 chips 56 times out of 100, betting 7 numbers!

Making this mistake is no different, in principle, to "calculating" that because there is a 99.9% chance of getting at least 1 black in 10 spins, then after 9 spins with no blacks the chance of a black on the next spin is 99.9%. This is of course, none other than the gambler's fallacy, but it may not be so easy to recognize it in more unusual or complex scenarios such as the probability of repeats.
« Last Edit: April 25, 2015, 11:27:16 AM by Slacker »


 

scepticus

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Re: A Common Error in Probability
« Reply #1 on: April 25, 2015, 02:24:13 PM »
In my years of reading on forums, this is the one error I see time and time again. It concerns confusing the probability of a series with that of a single.

For example, there is a famous example in probability called the "Birthday Problem" which states that in a room of 23 people, there is a 50% chance that at least two will share a birthday. Does this suggest a system for roulette? what is the probability that in a sequence of spins, you will get at least one repeat?

If you do the math, it turns out that in any 8 spin sequence, there is a roughly 56% chance that there will be at least one repeat in the sequence. No problem with that, but the error occurs when you make a statement like this:

"So it appears that after 7 consecutive numbers have appeared and none are repeats there is a 56% chance that one of the 7  will repeat on the next spin."

This is saying that there is a 56% chance that you will get a win when betting on 7 numbers! but any casino which offered the equivalent of such odds would soon be out of business. The mistake lies in assuming that you can apply the probability of the series (7 spins with at least one repeat) to that of a single outcome. But once the 7 spins have gone, probability applies to the next spin only, so the original probability is now meaningless. All you can do is bet the last 7 numbers and hope that one of the last 7 repeats. What is the chance of that?

The answer is 7/37, no more and no less. If you doubt this, assume that the last 7 numbers were 17,1,32,25,8,12,28. The chance that 17 will hit on the next spin (and so result in a repeat) is 1/37, the chance that 1 will repeat is again 1/37. Similarly for each of the others. Since these are mutually exclusive outcomes, we can add the results, which gives 7/37.

You can indeed make a system out of the knowledge that there is at least one repeat with probability 56% in the last 8 spins, but in order for the probability to remain valid, you have to place your bets from spin 1, not after spin 7. So on spin 1 you put one chip on the last outcome, on spin 2 add another chip to whatever just hit, and so on, until you get a repeat (a win). But in that case, your profit will vary according to when you get the repeat, assuming you do get it. 56 times out of 100 you will indeed get at least one, but what you cannot say is that you will win 29 chips 56 times out of 100, betting 7 numbers!

Making this mistake is no different, in principle, to "calculating" that because there is a 99.9% chance of getting at least 1 black in 10 spins, then after 9 spins with no blacks the chance of a black on the next spin is 99.9%. This is of course, none other than the gambler's fallacy, but it may not be so easy to recognize it in more unusual or complex scenarios such as the probability of repeats.

Aaaah ! Seems that you have accessed my website “ fergusleesroulette.co.uk “. and found my “ Birthday Method “.
The main  function of  roulette websites is to put forward ideas for discussion that REAL players have used -PROFITABLY.
I had used this method profitably but , as I said, I sometimes had to wait for a long time to have a bet. So not for me. The Long Run argument that critics use is invalid unless you can specify how long  your long run is.Furthermore, Probability Theory deals with Expectations and not the Certainties that our critics claim. It is a tool to be used and not the Holy Grail that our critics claim it to be.
So now that you have accessed my site would you be kind enough  to  programme  my Double Dozen strategy - which I use  profitably. I would like to know just when I  should stop betting it before losing not only my bankroll but all my profits from it .
I am pleased that we now have  a programmer who is able to programme any strategy. Thanks in anticipation.
 

dobbelsteen

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Re: A Common Error in Probability
« Reply #2 on: April 25, 2015, 02:40:23 PM »
Nobody can predict the outcome of the next spin.
The chance of a repeater is for the EC 50%. A sampl of 8 spins has 256 possible different sequences. Every sequence need 8 results. The number of all the outcomes is 256x8=2048 spins
One of the sequences has not a single repeater.
A 10 spins sample contents one sequence of 10 red on 1024 sequences.This is 0,098% .
 

Bayes

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Re: A Common Error in Probability
« Reply #3 on: April 25, 2015, 03:55:20 PM »
The Long Run argument that critics use is invalid unless you can specify how long  your long run is.Furthermore, Probability Theory deals with Expectations and not the Certainties that our critics claim. It is a tool to be used and not the Holy Grail that our critics claim it to be.

hmmm... I'm not sure what you mean by this, but it seems a little strange that you've used probability on your site to support a system but are now dismissing it because it turns out that the chances aren't as favorable as you thought they were.

Anyway, regarding your system, it's not easy to understand, so do you have any further examples or explanations here? How long have you been playing it (number of placed bets) and with what results?
 

scepticus

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Re: A Common Error in Probability
« Reply #4 on: April 25, 2015, 04:44:03 PM »
Hi Slacker,
 In the absence of zero there are 81 possibilities in 4 spins of the wheel when betting the dozens 3x3x3x3 = 81. Grouping these 81 into 9 groups of 9 ,  my Blocks - or Groups -guarantees 3 correct no matter the result .
Using Block 1 , which is
1 1 1 2 2 2 3 3 3
1 2 3 1 2 3 1 2 3
2 3 1 3 1 2 1 2 3
2 3 1 1 2 3 3 1 2 an example would be if the 4 spins are won by  numbers in the dozens 1 -1 - 3 - 2 it can be seen that 3 of these are in the first line of the block
.As will be seen we cannot actually bet the first 2 "spins " as we would need to bet all 3 each time for no profit but a loss should zero occur.Hence these need to be used as " virtual spins "and used only as a guide as to what to bet on spins " 3 " and "4".
4  combinations of 3 are needed to ensure 3 correct from 4 spins. These are 1-2-3 / 1-2-4 / 1-3-4 / and 2 /3 /4 and so is a 1 in 4 shot for a payoff of 7 / 2 and so ,  even allowing for the zero , gives an advantage to the player.
It may be thought that as there are nine combinations of spins 3 and 4 then it is really a 2 in 9 shot for a payoff of 7 /2 but  we finish when any 3 win so we cannot bet the 4th of a winning 1-2-3. Even if we did we would need to bet all 3 dozens which would be silly.
As the 3rd spin , if won , pays only 2 /1 we do not bet it but place 1 chip on each of the other 2 dozens and , if one of them wins  , put the resulting 3 chips on the 4th number.
Read and digest .
With  volatility and the need for virtual spins I think programming it
would be challenging - but then , I am not a programmer.The best of luck.

 

Bayes

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Re: A Common Error in Probability
« Reply #5 on: April 26, 2015, 02:14:24 PM »
Scepticus,

I'm none the wiser, I'm afraid. This is far too vague, computers are completely stupid and only do what they're told to do, no more and no less, so if I can't play your system manually (which I can't, at the moment), then I certainly won't be able to tell the computer how to do it.

If you could show how you play these 20 spins with step-by-step instructions, I might be able to get a handle on it. Thanks.

21
19
29
36
7
7
16
12
20
2
7
11
6
34
24
35
12
8
23
24

 

scepticus

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Re: A Common Error in Probability
« Reply #6 on: April 26, 2015, 11:32:32 PM »
Hi slacker. Thanks for your question .To keep things tidier  I have given my reply in Scep's Roulette Strategies.
 

palestis

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Re: A Common Error in Probability
« Reply #7 on: April 27, 2015, 03:08:11 AM »
In my years of reading on forums, this is the one error I see time and time again. It concerns confusing the probability of a series with that of a single.

You can indeed make a system out of the knowledge that there is at least one repeat with probability 56% in the last 8 spins, but in order for the probability to remain valid, you have to place your bets from spin 1, not after spin 7. So on spin 1 you put one chip on the last outcome, on spin 2 add another chip to whatever just hit, and so on, until you get a repeat (a win). But in that case, your profit will vary according to when you get the repeat, assuming you do get it. 56 times out of 100 you will indeed get at least one, but what you cannot say is that you will win 29 chips 56 times out of 100, betting 7 numbers!

Yes your logic complies with probability theory. But there is another force that enters the picture.
If you study many many numbers, you will find that this 56% or any other percentage, takes place MORE FREQUENTLY within a range determined by empirical research. In your example, will the repeat occur in the very 1st spin, or will it occur in the 8th spin? Very unlikely. Lets say after processing hundreds of thousands of numbers, you will find that the 56% remains true to its probability, however the 56% happens a lot more often within the range of the 3rd spin to the 7th spin. Meaning that if the 56% chance of a repeat stands, it is more likely to happen  between the 3rd and the 7th spin. If that's true, doesn't make more sense to bet this range, instead of the entire 8 spin range?
Secondly, who says that if you have to bet all 8 spins to enjoy the 56% chance, you can't bet $0.50 in the first 2 spins, then $10 from spin 3 to 7 and then stop?
Empirical research, is not against probability. It goes a further step to determine the range within which, the action is most likely to happen.
These are the issues that probability theory does not address. It can only assign values for individual events or series of events. The range of action is something only a player can determine, after extensive research. And that results in probabilities of occurrence  within a specific range. What's wrong with using these probabilities?
Police departments are on guard for driving violators. Y are they more vigilant between 1 am and 4 am? And far more relaxed between 9 am and 5 pm?
Range of occurrences is  part of every day life. Roulette is not an exception.
« Last Edit: April 27, 2015, 03:28:16 AM by palestis »
 

Real

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Re: A Common Error in Probability
« Reply #8 on: April 27, 2015, 03:21:38 AM »
Guys,

Again, you can't step outside of probability with your virtual bets and take advantage of bets that you believe are "due".    Again, gambler's fallacy.  Look it up.
 

palestis

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Re: A Common Error in Probability
« Reply #9 on: April 27, 2015, 03:35:55 AM »
Guys,

Again, you can't step outside of probability with your virtual bets and take advantage of bets that you believe are "due".    Again, gambler's fallacy.  Look it up.
How is the fact that the probability of 8 black in a row is [18/37]^8, outside probability? And how "what is due" fits in? We are talking about  at least one event happening within a series of events.
 

Bayes

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Re: A Common Error in Probability
« Reply #10 on: April 27, 2015, 07:47:39 AM »
Quote
Secondly, who says that if you have to bet all 8 spins to enjoy the 56% chance, you can't bet $0.50 in the first 2 spins, then $10 from spin 3 to 7 and then stop?

Of course you can do that, it's called a progression. My point was that you cannot take the probability of the series and apply it to the next spin.

Quote
These are the issues that probability theory does not address. It can only assign values for individual events or series of events.

Umm... there are ONLY individual events and series of events, they encompass the totality of possibilities, so what is it that probability theory can't deal with?

Quote
The range of action is something only a player can determine, after extensive research. And that results in probabilities of occurrence  within a specific range. What's wrong with using these probabilities?

There are not two kinds of probabilities - "theoretical" and "empirical". Assuming a fair wheel, all empirical probabilities will conform to the theoretical probabilities. I think the problem is that people don't understand that the word "theoretical" in this context doesn't mean a guess, or something that hasn't been verified by experience. In fact it means the opposite.

In science, the term "theory" refers to "a well-substantiated explanation of some aspect of the natural world, based on a body of facts that have been repeatedly confirmed through observation and experiment."[14][15] Theories must also meet further requirements, such as the ability to make falsifiable predictions with consistent accuracy across a broad area of scientific inquiry, and production of strong evidence in favor of the theory from multiple independent sources.
 

dobbelsteen

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Re: A Common Error in Probability
« Reply #11 on: April 27, 2015, 09:15:25 AM »
I do not understand what you mean with the error in propability.
From my point of view the propability theory fails for a short run event.The propability theory is very interested for researching large events. A roulette player plays always small events.
 

palestis

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Re: A Common Error in Probability
« Reply #12 on: April 27, 2015, 09:25:04 AM »
Quote
Secondly, who says that if you have to bet all 8 spins to enjoy the 56% chance, you can't bet $0.50 in the first 2 spins, then $10 from spin 3 to 7 and then stop?

Of course you can do that, it's called a progression. My point was that you cannot take the probability of the series and apply it to the next spin.

Quote
These are the issues that probability theory does not address. It can only assign values for individual events or series of events.

Umm... there are ONLY individual events and series of events, they encompass the totality of possibilities, so what is it that probability theory can't deal with?
The point is that the probability of series stands, whether you are in the second spin or third or fifth. As long as you start the betting process form spin one. When you deal with a series of bets, there is only one probability from the beginning until what you are aiming for  happens, or till the end of the series. Is that understood?
The value doesn't change if you missed the first 2 or 3 spins in an 8 preplanned spin series.
If you missed one or a few spins, that doesn't mean that the probability of series is invalid, and it only counts what happens after the missed spins. Otherwise it would make no sense to have probability of series. So my question in your example is this. Is the probability of a repeat in 8 spins still 56%, after  missing the first 6 spins?  (Yes the betting process started form spin 1). The probability of series specifies a specific percentage of at least one success in a preplanned 8 series of bets. That percentage does not and should not change just because the first 6 spins missed the target.
Of course seeing 6 missed spins somewhere on a score board, is not the same as being part of the betting process from spin 1. Ready made virtual losses do not count (up to a certain extend). The player  HAS TO BE in the betting process from the start. The bet amounts is another issue,  and that's where a construction of a good  system starts.

 

palestis

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Re: A Common Error in Probability
« Reply #13 on: April 27, 2015, 09:28:31 AM »
I do not understand what you mean with the error in propability.
From my point of view the propability theory fails for a short run event.The propability theory is very interested for researching large events. A roulette player plays always small events.
Yup. I forgot to mention that. Though probability is correct in large data, when it comes to roulette and short data probability is always WRONG. And that's what a smart player can take advantage of.
 

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Re: A Common Error in Probability
« Reply #14 on: April 27, 2015, 11:21:03 AM »
Quote
when it comes to roulette and short data probability is always WRONG. And that's what a smart player can take advantage of.

No, it's just your comprehension of it that is wrong.  And no, the smart player can't consistently  take advantage of several short run plays, since the house payoff is short of what probability dictates fair.

-Real