Probability Theory and CTs

[Nick Terry]

I've seen a few of the veteran posters on here refer to probability theory from time to time, as a means of illustrating how implausible conspiracy theories are. At the same time, I also recall seeing someone use probability theory to demonstrate the reliability of witnesses.

apropos the first, pomeroo posted the following a few months back:

Suppose that instead of thousands of people knowing the secret, there are only one hundred. Suppose further that these hundred people are extraordinarily good at keeping their mouths shut. Assign them an average probability of .9 (a typical human's might be .5) of never spilling the beans. This conspiracy--much, much smaller than yours--has a 99.97% probability of letting the cat out of the bag (use a calculator to raise .9 to the hundredth power).

The gigantic network of perps, accomplices, and coerced innocents posited by conspiracy liars would let the secret slip far more often, 99.99999...% of the time. The math is childishly easy. You can let everyone involved clam up with a probability of .99--they're all James Bonds and G. Gordon Liddys--and your conspiracy will still unravel with near-certainty.

The above example makes perfect sense even to a mathematical layman such as myself.

Regarding the second possible use of probability theory, it strikes me that this is an already built-in feature of Western legal theory, and everyday thinking. This goes right back to the Judaic and Roman law principle that two witnesses were sufficient evidence to regard something as proven.

One can illustrate the two-witness principle fairly simply. If one assigns a 50/50 chance that any witness is telling the truth, then if there are two who say the same thing, the probability that they are both telling the truth rises to 75%.

This is an important threshold, since common law and current US law has several standards of proof. The first is the balance of probabilities, i.e. 51%, the second is clear and convincing evidence, which one legal theorist has assigned at 75%, and the third is beyond reasonable doubt.

Lawyers are extremely reluctant to assign a statistical probability to the last level, since to assign a figure of say, 90%, would imply that 10% of the time it would still be reasonable to doubt. Philosophers, however, argue that there can be no certainty, so clearly beyond reasonable doubt must be less than 100%. It is a reversible error for a judge to inform a jury that if they are 90% certain then that is 'beyond reasonable doubt'. In fact there are serious contradictions within the US court system as to whether it is permissible to explain what 'beyond reasonable doubt' means at all. Different states have different precedents regarding this. Until recently, the phrase 'moral certainty' was often used, but apparently that's now too confusing for juries so it can't necessarily be used.

But let's leave the philosophical and legal discussions to one side for the moment. I mention them because they tie in with 'everyday', common-sense usages, and because they are routinely abused by CTists.


So, on to the questions I'd like to throw out for discussion. The first is how to more precisely model the use of probability theory regarding witnesses. The second is how to guard against the abuse of probability theory by CTists, and to guard against potential counterattacks.

1. Regarding the first use, let me draw on an example from my field, history, and my debunking interest, combating Holocaust denial. Holocaust deniers are notoriously sceptical that any witness may be telling the truth. In fact they flatly state as a matter of course that they are all liars.

Therefore, let us entertain the possibility that this might be so, and assign a 90% chance (as in pomeroo's example above) that any one witness might be a liar.

Am I correct in deducing from pomeroo's demonstration of probability theory that if there were 100 witnesses each with a 90% chance of being a liar, that there is in fact a 99.97% chance that they are all telling the truth?

On the face of it, it seems counterintuitive but when this has been used before on this forum in a similar fashion, it made sense. The common-sense interpretation would be, surely, there is a 90% chance that they are lying, therefore they are all lying.

However, it surely also follows that if 100 liars are lying, and we assign a 90% chance that they can keep silent about their conspiracy, as in pomeroo's example, that the conspiracy will be let out of the bag.

Does this follow? It strikes me that if it does, then probability theory is a useful double-edged tool. It would surely demonstrate both that a large group of witnesses are telling the truth, and also demonstrate that for all of them to be lying is wholly improbable.

2. Regarding the potential abuses by CTists, how would one combat a potential abuse? One could take the example of cherrypicked quotes from eyewitnesses (actually, 'earwitnesses') which could be construed as evidence in support of controlled-demolition on 9/11.

I may be worrying unnecessarily here, since it would be surely unlikely that any CTist would concede that there is any chance he might be wrong, and assign a probability to his beliefs.

But let us assume there is a particularly dishonest and better-educated specimen of CTist who wants to abuse probability theory in order to make a quasi-rhetorical point.

What kinds of abuse could the CTist try on, and how would one counter them?

3. How can we use probability theory in relation to cherry-picking? If only a certain number of witnesses claim a certain anomaly, then they form a tiny percentage of a much larger group of witnesses, surely. For example, only a few first-responders claim to have heard noises 'like an explosion'.

How would one model this aspect?

Again, there are some examples in my own debunking field. Some Holocaust deniers claim on the basis of a few allegations that all key witnesses were coerced into giving false testimony. They have to, because otherwise their beliefs are simply refuted.

Now, this claim is surely vulnerable to the standard probability-theory conspiracy-busting argument put forward by pomeroo. The chance that 100 torture victims all kept silent is close to nil.

However, the claim is surely spurious to begin with, since if one had, say, six allegations out of 206 interrogated war criminals, then simple percentages suggest that only 2.9% of the set of interrogated war criminals even made allegations regarding coercion, let alone could be proven to have been coerced.

How would one then quantify the probability that all had been tortured? I ask because it would surely be quite a leap to go from six examples out of 206 (to keep with the figures; they roughly correspond to the number indicted at the Nuremberg and successor trials) to infer that all had been coerced.

What would the probability be? Surely extremely low to non-existent, even assuming a high level of credulity to this particular conspiracy theory.



Basically, help! I'm a historian, and fine with addition, subtraction, percentages and historical statistics in the general sense, but never did the full statistical-mathematical training.

I sense that there are some very useful points to be made with probability theory, but I want to know if there are any pitfalls. The thing I like about the examples that have been used on this forum is that one can assign different standards of proof, from the everyday to the ultra-sceptical, and in many cases demonstrate that what might seem 'obvious' to a CTist is in fact drastically improbable.

 

[fezzic]

Am I correct in deducing from pomeroo's demonstration of probability theory that if there were 100 witnesses each with a 90% chance of being a liar, that there is in fact a 99.97% chance that they are all telling the truth?

Actually there is a 0.003% chance that all 100 lie and a 99.997% chance that one will tell the truth. As for all 100 telling the truth, my calculator does not go that far, it is such a small number.

So to follow the line, even with 90% chance of each of the 100 participants (with full knowledge) keeping the secret, the chances of one of them telling the truth is virtually certain. That's the one that security has to find.

 

[Nick Terry]

Actually there is a 0.003% chance that all 100 lie and a 99.997% chance that one will tell the truth. As for all 100 telling the truth, my calculator does not go that far, it is such a small number.

So to follow the line, even with 90% chance of each of the 100 participants (with full knowledge) keeping the secret, the chances of one of them telling the truth is virtually certain. That's the one that security has to find.

OK, I see the distinction. However, the issue is 100 people all saying the same thing. If you say there's a 0.003% chance that all 100 are liars, and a 99.997% chance that one is telling the truth, then if the one that is telling the truth is saying the same thing as the other 99 people, then they are all surely telling the truth. That is even if one assigns a scepticism level of 90% (in this example).

The example is simply to illustrate the absurdity of excessive scepticism. I mentioned already the old Judaic and Roman law standard that two witnesses were sufficient evidence to convict. The two-witness rule still applies in journalism, and is everyday thinking. If one person gives me directions to a destination then they might be misleading me, but if I approach two people separately and they both give the same route, then it is unlikely they are both lying.

In a normal context, if one is presented with 5, 10, 20, 25, 30 sources saying the same thing, it becomes harder and harder to justify disbelief. One could theoretically quantify that with probability theory.

However, in a CT context, one is faced with unwarranted certainty masquerading as extreme scepticism. Thus the assignment of a .9 probability that something may not be the case, just to bend over backwards to give the benefit of the doubt.

Note, I think one of the issues is how one defines 'all saying the same thing'. It is common to distinguish between a core narrative in a testimony, an essential account, and the peripheral details. CTists love to seize on the variable details and argue from anomaly. Again, maybe probability theory can be used to illustrate this. The chances that 100 people will all tell the story in exactly the same way down to the last detail are surely vanishingly low. The chance that 100 people will tell the essential same story are surely incredibly high. The chance that a contradiction in minor details between a group of witnesses signifies that all are mistaken/lying is surely exceedingly low.

 

[Drudgewire]

There's a 100% chance reading any more of that will make my brain hurt.

 

[Nick Terry]

There's a 100% chance reading any more of that will make my brain hurt.

Mine too, which is why I hope our more mathematically-educated members can weigh in sooner or later...

 

[Architect]

I mentioned already the old Judaic and Roman law standard that two witnesses were sufficient evidence to convict. The two-witness rule still applies in journalism, and is everyday thinking.
quote]

And in Scots Law, which is modelled on Roman Law.

http://en.wikipedia.org/wiki/Scots_law

Sorry for the derail.

 

[TerryUK]

... the issue is 100 people all saying the same thing.

If 100 people are saying the same thing, then your arbitrary .9 value becomes irrelevant.

 

[Nick Terry]

If 100 people are saying the same thing, then your arbitrary .9 value becomes irrelevant.

No, it doesn't. Let me illustrate with a more concrete, less sensitive example. 100 convicts, all members of the mafia, uniformly claim that a particular prison guard beat them. Convicts have a low level of trustworthiness and the mafia had a high level of solidarity, i.e. a low level of likelihood that they will brag. It may be arbitrary to assign both as .9, but the consequences become clear.

If the allegation is false, then even a high-solidarity group will have someone who spills the beans and reveals that the allegation is an orchestrated conspiracy. Someone among the 100 mafia convicts would brag and be overheard.

If the allegation is true, then no one brags.

 

[Architect]

You know, I suspect that Casinos make a whole rake of cash out of this kind of issue. I'd ask a bookie....

 

[Brainster]

The smarter type of CT will claim that witnesses have come forward: Kevin McPadden, John Schroeder, Indira Singh, Sibel Edmonds, etc. This throws the probabilities out the window, because the numbers don't go stratospheric. If you flip a coin 16 times and it comes up heads everytime, the odds against that are very high. But it you flip that same coin 16 times and it comes up 14 times heads and two times tails, the odds against that are not anywhere near as high.

Abuse of probability is quite common in the 9-11 "Truth" movement. A common mistake is to apply percentages of likelihood to various parts of 9-11 and multiply them together to obtain the likelihood of 9-11 happening. Here's an example:

The success of the 9/11/01 attack depended on each of the following five events happening as planned:

* Four jetliners were taken over with no effective resistance.
* Three of the four jetliners were flown into small targets, with the Pentagon strike involving extreme aerobatic maneuvers.
* The air defense network froze for over an hour while the attack unfolded.
* The towers self-destructed in a manner never before seen in any structure.

There's a rather obvious flaw in this analysis, beyond the fact that you cannot apply probabilities to a past event to "prove" that the event could not have happened, and that is the bit about "three of the four jetliners were flown into small targets..."

Of course, the plan was for four jetliners to be flown into targets (it's rather incongruous to call the WTC buildings and the Pentagon "small"), not three.

 

[Nick Terry]

The smarter type of CT will claim that witnesses have come forward: Kevin McPadden, John Schroeder, Indira Singh, Sibel Edmonds, etc. This throws the probabilities out the window, because the numbers don't go stratospheric. If you flip a coin 16 times and it comes up heads everytime, the odds against that are very high. But it you flip that same coin 16 times and it comes up 14 times heads and two times tails, the odds against that are not anywhere near as high.

Abuse of probability is quite common in the 9-11 "Truth" movement. A common mistake is to apply percentages of likelihood to various parts of 9-11 and multiply them together to obtain the likelihood of 9-11 happening. Here's an example:

There's a rather obvious flaw in this analysis, beyond the fact that you cannot apply probabilities to a past event to "prove" that the event could not have happened, and that is the bit about "three of the four jetliners were flown into small targets..."

Of course, the plan was for four jetliners to be flown into targets (it's rather incongruous to call the WTC buildings and the Pentagon "small"), not three.

Apropos the second 'probability analysis', the obvious rebuttal is that there WAS a series of past events which belong together in a family group, namely foiled terrorist plots. Didn't Richard Clarke give some kind of number for how many supposedly foiled plot there were prior to 9/11? We have obviously had such figures for after 9/11.

Viewed in this context, 9/11 appears much less improbable than if one isolates the event and concentrates solely upon it. Naturally, CTists seem disinclined to accept that there ever may have been other terrorist plots, foiled or successful.

As the saying goes, the terrorists only need to be lucky once; which they were on 9/11, and even then as you rightly point out, 25% of the attacks 'failed'.


Regarding the first point, the probabilities may not be as stratospheric but they remain, surely, very high. That is why one should bear in mind the various standards of proof. Also, it depends absolutely on what one considers to be a 'whistleblower'. No one has come forward to say 'I planted explosives in the Twin Towers'.

 

[TerryUK]

... beyond the fact that you cannot apply probabilities to a past event to "prove" that the event could not have happened...

Don't they do that a lot in criminal trials ?

 

[TerryUK]

If the allegation is false, then even a high-solidarity group will have someone who spills the beans and reveals that the allegation is an orchestrated conspiracy.

This is where things are a little different from the roulette wheels and coin tossing etc.

You say "even a high-solidarity group" will have someone who spills the beans...

How do you decide just how likely any one of them is to talk?

This value is important, especially within a small group.

At some point, as the number in your group becomes bigger, this value -the likelyhood that any one in the group would talk- becomes much less important. (as far as being confident that someone would talk )

e.g. 10 in the group, vs. 1,000 in the group.

With 1,000 in the group, it doesn't matter if the likelyhood of an individual 'talking' is only .01 (1%) -- you would be very likely to have someone talk...

But with only 10 in the group, you'd only have a .1 (10%) probability that someone would spill the beans.

In other words, regarding 9/11, a 'small-ish' group of people with a very strong disincentive to talk would be quite consistent with a situation fitting the the 9/11 CT ( LIHOP or MIHOP )

 

[TriskettheKid]

The smarter type of CT will claim that witnesses have come forward: Kevin McPadden, John Schroeder, Indira Singh, Sibel Edmonds, etc. This throws the probabilities out the window, because the numbers don't go stratospheric. If you flip a coin 16 times and it comes up heads everytime, the odds against that are very high. But it you flip that same coin 16 times and it comes up 14 times heads and two times tails, the odds against that are not anywhere near as high.

Funny story, in 9th grade my science fair project dealt with coin flips. I ended up flipping 100 quarters approximately 30,000 times (though only about 20,000 flips met my control variables).

As such, I did research on probability and this wonderful thing called Pascal's Triangle.

Now, the odds of you getting 16 heads in a row is roughly 1 in 65,536. The odds of getting 14 heads and 2 tails in 16 tosses is equivalent to the triangular number of the 16th row of Pascal's Triangle over 65,536. In other words, it's equivalent to 105 in 65,536. So just getting 2 tails makes the result 105x more likely, or an event that would happen roughly 1 in 624 times.

This is where things are a little different from the roulette wheels and coin tossing etc.

You say "even a high-solidarity group" will have someone who spills the beans...

How do you decide just how likely any one of them is to talk?

This value is important, especially within a small group.

At some point, as the number in your group becomes bigger, this value -the likelyhood that any one in the group would talk- becomes much less important. (as far as being confident that someone would talk )

e.g. 10 in the group, vs. 1,000 in the group.

With 1,000 in the group, it doesn't matter if the likelyhood of an individual 'talking' is only .01 (1%) -- you would be very likely to have someone talk...

But with only 10 in the group, you'd only have a .1 (10%) probability that someone would spill the beans.

In other words, regarding 9/11, a 'small-ish' group of people with a very strong disincentive to talk would be quite consistent with a situation fitting the the 9/11 CT ( LIHOP or MIHOP )

Alright, let's give each person a .999 chance of not talking in a group of ten.

The odds, then, that someone will not talk is .9900448802097482, in other words, there's a 1 in 100 chance that someone will have talked.

But even an increase of one person will drastically change the numbers.

For a group of 15 people, all with a .999 likelihood of not talking, you have a roughly 1.5% chance of someone talking (.985 of no one talking).

Group of 20 people? You have a roughly 2% chance of someone talking (.02) and a 98% chance of no one talking (.98).

Group of 30? You have a 3 in 100 chance of someone talking.

50? Yup, down to a 95% chance of no one talking.

It turns out that once you get to around 4000 people involved, each with a .999 chance of not talking, you get a 98% chance that someone will talk.

And that's if we're using an incredibly conservative estimate of .999, a model in which everyone involved has no problem with going through with the plan.

 

[jhunter1163]

Someone on this forum had a sig at one point that said "Watergate Ratio: For every 12 conspirators, there is one Mark Felt (Deep Throat)", or something to that effect. I tend to agree. There are many, many reasons that secrets leak out; conscience, moral outrage, money, or just plain "I know something you don't know, neener neener neener."

Probability doesn't really even have to enter into it. It's human nature not to keep secrets. Maybe one or a few people could fight down the urge to talk, but it's just silly to bet against human nature when hundreds or thousands of humans are involved.

 

[Mel Odious]

From Nick Terry's original post:

One can illustrate the two-witness principle fairly simply. If one assigns a 50/50 chance that any witness is telling the truth, then if there are two who say the same thing, the probability that they are both telling the truth rises to 75%.

Someone please correct me if I'm wrong about this, but I believe that the probability that both witnesses are telling the truth (if they say the same thing) is still 50/50.

Let's compare your scenario to one where we flip two coins, and consider only the times when both come up heads or both come up tails. Obviously, we would expect double-heads half the time and double-tails the other. We certainly would not expect three double-heads per four trials over the long run. Similarly, if any witness is equally likely to give true or false testimony, each group of two witnesses giving true testimony would be balanced out by two witnesses giving false testimony, over the long run.

Does that seem right, everybody?

 

[Dave Rogers]

Someone please correct me if I'm wrong about this, but I believe that the probability that both witnesses are telling the truth (if they say the same thing) is still 50/50.

I think you're wrong here, and the reason you're wrong is that a coin toss is a poor analogy. If both witnesses say the same thing, there are three possibilities: (1) They are both telling the truth, (2) they have got together and agreed upon a lie, and (3) they have at random come up with the same lie. (3) is extremely unlikely because the set of untrue scenarios is very large, so if (2) can be eliminated, (1) is by far the most likely scenario. On the other hand, if witnesses disagree, there are many possibilities, including poor recollection, one or both witnesses lying, and individual witnesses having seen different subsets of events.

The point is that in a coin toss there are only two possible results to choose from. For witness statements, the set of possible lies is far greater than the set of possible subsets of the truth that each witness will know. The probability of a witness lying, therefore, is much greater than the probability of a witness telling a specific lie that agrees with another lie, therefore it's rather easier to eliminate untrue testimony simply by looking for inexplicable inconsistencies.

Dave

 

[Nick Terry]

This is where things are a little different from the roulette wheels and coin tossing etc.

You say "even a high-solidarity group" will have someone who spills the beans...

How do you decide just how likely any one of them is to talk?

This value is important, especially within a small group.

At some point, as the number in your group becomes bigger, this value -the likelyhood that any one in the group would talk- becomes much less important. (as far as being confident that someone would talk )

e.g. 10 in the group, vs. 1,000 in the group.

With 1,000 in the group, it doesn't matter if the likelyhood of an individual 'talking' is only .01 (1%) -- you would be very likely to have someone talk...

But with only 10 in the group, you'd only have a .1 (10%) probability that someone would spill the beans.

In other words, regarding 9/11, a 'small-ish' group of people with a very strong disincentive to talk would be quite consistent with a situation fitting the the 9/11 CT ( LIHOP or MIHOP )

As the saying goes, "three people can keep a secret if two are dead".

jhunter1163 put it pretty well:

Someone on this forum had a sig at one point that said "Watergate Ratio: For every 12 conspirators, there is one Mark Felt (Deep Throat)", or something to that effect. I tend to agree. There are many, many reasons that secrets leak out; conscience, moral outrage, money, or just plain "I know something you don't know, neener neener neener."

Probability doesn't really even have to enter into it. It's human nature not to keep secrets. Maybe one or a few people could fight down the urge to talk, but it's just silly to bet against human nature when hundreds or thousands of humans are involved.

In the case of 9/11, a conspiracy would not simply be vulnerable to secrets leaking out from a decision-making group (which might well consist only of ten people), but from those tasked/ordered/paid to carry out the desired action, and from those in a position to make direct, meaningful observations of the results. This rapidly escalates the numbers and widens the circle of those who must keep silent.

One can certainly postulate that an inner circle of something so momentous would keep silent. The probability that this solidarity will maintain decreases exponentially as the circle is widened.


Dave Rogers:

I think you're wrong here, and the reason you're wrong is that a coin toss is a poor analogy. If both witnesses say the same thing, there are three possibilities: (1) They are both telling the truth, (2) they have got together and agreed upon a lie, and (3) they have at random come up with the same lie. (3) is extremely unlikely because the set of untrue scenarios is very large, so if (2) can be eliminated, (1) is by far the most likely scenario. On the other hand, if witnesses disagree, there are many possibilities, including poor recollection, one or both witnesses lying, and individual witnesses having seen different subsets of events.

The corollary with disagreement is something poorly understood by CTs.

The point is that in a coin toss there are only two possible results to choose from. For witness statements, the set of possible lies is far greater than the set of possible subsets of the truth that each witness will know. The probability of a witness lying, therefore, is much greater than the probability of a witness telling a specific lie that agrees with another lie, therefore it's rather easier to eliminate untrue testimony simply by looking for inexplicable inconsistencies.

The keyword being inexplicable, and human perception being what it is, the chances of outward inconsistencies entering into any one group of testimony (for example, 9/11 witnesses) rise to a certainty.

That's why cherry-picked quotes about hearing sounds 'like an explosion' are insufficient evidence upon which to base a controlled-demolition theory.

 

[maxpower1227]

IOne can illustrate the two-witness principle fairly simply. If one assigns a 50/50 chance that any witness is telling the truth, then if there are two who say the same thing, the probability that they are both telling the truth rises to 75%.

This is nitpicking, and doesn't challenge the overall premise of the thread, but I don't think that is correct. If your two witnesses give testimony independent of the other, the odds are 75% that at least one of them tells the truth; the probability that both tell the truth is 25%. If you enforce the condition that both must tell the same story, they are either both telling the truth or both lying: probability 50% for each.

 

[ElMondoHummus]

Guys, pardon me as a non-math expert - who, true to his numerical idiocy stayed firmly entrenched in the qualitative sciences as an undergrad - but I have to ask: Aren't we misapplying things? Probablity is a method of analyzing random, non-deterministic events. Human behavior has some rather strong deterministic aspects, such as motive, belief systems, prior experiences, etc.

I don't think probability can really be applied to witness tesimony or other aspects of analyzing human responses like we've been doing here. Then again, IANAME (I Am Not A Math Expert) so I'm very subject to correction by a trained mathematician. Or anyone else with a stronger grasp of the concepts than I have. But, that is how I understand this.