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Simple - Isn't it?

If in other sciences we should arrive at certainty without doubt and truth without error,
it behooves us to place the foundations of knowledge in mathematics.
Opus Majus, bk. I, ch .4   Roger Bacon  c.1214 - c.1294


 The simple believe everything, but the clever consider their steps.
Proverbs 14:15
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Victor Trombettas mentions (http://usread.com/flight587/Reliability/Reliability.html) a New York Times article published on June 23, 2002 related to the Flight 587 investigation titled "For Air Crash Detectives, Seeing Isn't Believing" making the case that in addition to offering little if any useful information, eyewitnesses offer such contradictory accounts, as to render them useless.  Since TWA Flight 800, it's long been known by those close to the NTSB, those who have read their TWA Flight 800 Final Report, and those who listened to the NTSB at Press Conferences after Flight 587 crashed in Belle Harbor, that witnesses are viewed by the NTSB as "notoriously unreliable".
 
On June 4th the NTSB released statistics categorizing the 349 witness statements they had received from witnesses to the crash of Flight 587.  52% specifically reported seeing a fire while the plane was in the air, with the fuselage being the most often cited location (22%). Other areas cited as a fire location were the left engine, the right engine or an unspecified engine, and the left wing, the right wing or an unspecified wing. In other words, 181 people saw a fire while the plane was in the air. 168 did not say they saw a fire.
 
In a December 15, 1996 article in the NY Times entitled "Many Answers in Crash Except the One That Counts" it is stated that: "safety board officials said their leading theory of the TWA 800 crash is that the fumes were ignited by a spark of static electricity created by fuel leaking into the tank. But they offered no evidence to support that theory and, in fact, they could not rule out a bomb or missile as the cause of the explosion -- two other possibilities for which there is no evidence."  
 
755 eyewitnesses to the TWA 800 disaster were interviewed by the NTSB/FBI and their testimonies were dismissed as contradictory, or influenced by past events.

March 21, 2000  CNN.com
Federal Investigators' Comments: The witnesses were on land, sea and air. Some were surfers. Many may have been influenced by news media reports about the explosion.

March 21, 2000  CNN.com
NTSB comment in its Witness Group Study report:  FBI witness interviewing was focused on the possibility that a missile had been used against the accident airplane. This focus may have resulted in bias on the part of some of the interviewers. The NTSB witness reports conclude that the cause of the explosion cannot be determined through eyewitness accounts alone.

In case the "FBI biased them" explanation might not be believed the NTSB considered floating an idea that the eyewitnesses were all suffering from "perceived memories". Elizabeth Loftus of the University of Washington, perhaps the world's greatest expert in manipulation of perceived memories, was scheduled by the NTSB to appear at the Baltimore hearings to help in the presentation of witness "perceptions" of the TWA Flight 800 disaster. Her presentation was canceled by joint FBI/NTSB agreement.  Tom Shoemaker asked Dr. Loftus: "Can you please share with me what your presentation was going to include?"   Dr. Loftus replied, "Basically, to address the question of why people thought they saw missiles when there weren't any."  Apparently the NTSB supplied Dr. Loftus with information proving to her that no missile(s) were seen by eyewitnesses, regardless of what they thought they saw on July 17, 1996.  

Of the 755 eyewitnesses interviewed by the NTSB/FBI let us assume that only 40 of them are "credible". Furthermore, let us assume that the observational capability of each of these TWA 800 eyewitnesses is so poor that he/she can report an observation correctly in only one out of five observations. There is then only a 20 percent probability that an event reported by such a witness would have actually taken place as described. With two of these witnesses independently describing the same event, the probability rises to 36 percent and with ten witnesses it rises to 89.26 percent. With forty such independent and similarly "unreliable" witnesses the probability rises to 99.99 percent that the event reported did in fact take place. This is as close to certainty as one can come mathematically. Further, even if one goes further and assumes that the reliability of these 40 eyewitnesses to TWA 800's destruction was so poor that each of them could report an observation correctly only one time out of ten, the probability that the event did indeed take place as described by the witnesses is still 98.52 percent.

The equation is as follows:

Probability of Event (P) = [1-(U^N) ] X 100
Percent Probability (P)  equals one minus the Unreliability (U) raised to the power of the Number of Participants (N) all multiplied by one hundred.

For example: If you give a coin to one person and ask them to flip it and you hope to see a "head" what is the probability that you will get it?

Answer 50%.

What if we give two people a coin each and ask both to flip their coins and we hope to have one "head" show up, the probability that we will get it is now 75%.

With three people the chances that we can get at least one head is 87.5% and with four people it is 93.75%.

Another way to look at these probability outcomes is as follows:  

Assume that one has identical pieces of equipment which need to operate for a set period of time but each can do so with only 50% reliability.

How many pieces of equipment do we need to operate simultaneously to ensure with 99.9% probability that we will get the job done?    

Answer 10.

Now if a witness is "right" in an observation only 10% of the time (or another way of putting this is that the witness is unreliable in 90% of his observations) then if one had 50 such witnesses what is the probability that what they said happened, actually happened?

The solution is shown below:

Accuracy of Witness Groups at Ten Percent Reliability

No of Witnesses                          Probability that Event Happened

           1                                             10.00 % (N=1 and U=0.9)

           5                                             40.95 % (N=5 and U=0.9)

          15                                             79.41 % (N=15 and U=0.9)

          20                                              87.84 % (N=20 and U=0.9)

          35                                              97.50 % (N=35 and U=0.9)

          50                                              99.48 % (N=50 and U=0.9)

Another example:

Suppose an Eastbound red car collides with a Southbound blue car at an intersection controlled by green-yellow-red traffic signals on all 4 corners. There are 100 witnesses. Fifty of the witnesses contend that the signals controlling Eastbound traffic were green when the red car entered the intersection and the other 50 witnesses contend that the traffic signals for Southbound traffic were green when the blue car entered the intersection. Both drivers were alone and both were killed.  Which driver ran the red light?

100% of the witnesses contend that one of the cars entered the intersection when the traffic light controlling it's direction of travel was green.

Was it the red car?

Was it the blue car?

1) First we must assume that the 100 witnesses are all independent i.e. they did not discuss the crash among themselves.

2) They are each 10% reliable as the table above assumes i.e. there is only a 1 in 10 probabality that any one of them saw the correct color of the lights.

Then you have a 99. 48 % probability that both sets of 50 eyewitnesses were in fact correct.

So how does one rationalize this?

What it means is that I can come to fully informed conclusion that there is a 99.48% probability that the traffic lights were indeed showing green in both directions at the same time.

We need no further analysis of the eyewitnesses by the police or their explanation of what they think happened.

We need to go and check out the traffic lights!

With the TWA 800 eyewitnesses we need no further explanation from the FBI, the NTSB, or the CIA of what they think happened.

We need to go and find the people who fired the missiles!

Simple. Isn't it?