## JFK Assassination: Mathematical Proof of a Grassy Knoll Shooter

05 Feb

JFK Assassination: Mathematical Proof of a Grassy Knoll Shooter

Richard Charnin
Feb 5, 2014
Updated: June 8, 2014

Click Reclaiming Science:The JFK Conspiracy to look inside the book.

JFK Blog Posts
JFK Calc Spreadsheet Database
Tables and Graphs

Dealey Plaza witness survey data is in the JFKCalc spreadsheet: https://docs.google.com/spreadsheet/ccc?key=0AjAk1JUWDMyRdDFSU3NVd29xWWNyekd2X1ZJYllKTnc#gid=65

This article discusses the importance of independent, corroborating witness evidence in a court of law. http://www.spmlaw.ca/jfk/shot_pattern_evidence.pdf

The General Problem
Given:
1) n witnesses claim they heard, saw or smelled shots from the Grassy Knoll
2) P is a probability estimate that a given witness would be correct
3) Witness observations are independent events
Then:
4) The probability PM that a given witness is MISTAKEN: PM = 1 – P
5) The probability PA that ALL n witnesses are MISTAKEN: PA = PM^n

This is an update to the original post which is referenced in JFKFacts.org: http://jfkfacts.org/assassination/is-there-mathematical-proof-of-a-grassy-knoll-shooter/#more-12235

Researcher Harold Feldman wrote that of 121 eyewitnesses: 51 (42%) said shots came from the Grassy Knoll area, 32 from the TSBD, and 38 had no opinion. http://spot.acorn.net/jfkplace/09/fp.back_issues/12th_Issue/51_wits.html

Given P = 0.42 is probability of a witness being correct in stating that shots came from the Knoll, then the probability PM = 0.58 = 1-.42 that the witness was mistaken.The joint probability PA that ALL 51 witnesses were mistaken and there was NOT a Grassy Knoll shooter is 0.58 to the 51st power.
PA = 0.58^51 = 8.6E-13 = 0.000000000000861 or of 1 in 1,161,909,568,739 or 1 in 1.16 trillion.

Therefore, the probability PS that there was a Grassy Knoll shooter is PS = 1-PA = 0.999999999999139

If we exclude the 38 witnesses who had no opinion as to the origin of the shots, then 51 of 83 (62%) said shots came from the Grassy Knoll and there is a 38% probability that a given witness would be mistaken. The probability that ALL 51 witnesses would be mistaken is much lower: PA = 0.38^51 = 3.71E-22
PA= 1 in 2,697,966,622,402,536,000,000 or 1 in 2.7 billion trillion!

Of course, if just ONE witness was correct in observing a shot from the Grassy Knoll, then all 51 were correct. Therefore, the Warren Commission’s conclusion that Lee Harvey Oswald was a Lone Gunman shooting from the Texas Book Depository was bogus.

Summary of Surveys
``` Survey..........Feldman McAdams Galanor Adjusted ```
``` Total Asked............121 241 216 223 Total Opinions..........83 100 110 133 Grassy Knoll............51 35 52 84 Book Depository.........32 61 48 36 Both TB and GK...........0 2 5 9 Other locations..........0 2 4 4 No opinion..............38 69 37 36 Not asked................0 72 70 54```

``` ```

```GK % of Opinions........61% 35% 48% 62% TB % of Opinions........39% 61% 44% 28% Probability ALL GK witnesses mistaken: Probability............E-22 E-07 E-17 E-43 ```

Of 241 total witnesses, 100 had an opinion on the origin of the shots. Sixty-one (61) said the TSBD and just 35 said the Grassy Knoll. These numbers vary greatly from the other surveys.

Stewart Galanor’s “216 Witnesses” http://www.history-matters.com/analysis/witness/Sort216Witness.htm
Of the 216 interviewed, 52 heard at least one shot from the knoll, 48 from the TSBD, 5 from both locations, 4 elsewhere. The probability PA that 52 witnesses who said shots were fired from the Grassy Knoll would ALL be MISTAKEN is E-17 (1 in 100,000 trillion).

A total of 224 witnesses were asked where the shots came from. Of the 133 who had an opinion, 84 said the Grassy Knoll; 36 said the TSBD; 9 both locations; 4 said elsewhere, 37 had no opinion.

The probability analysis was posted in jfkfacts.org and elicited the following comments: http://jfkfacts.org/assassination/response-to-charnin-the-grassy-knoll-probability-problem/#comment-342727

Acoustic Evidence
In 1978 at the HSCA, an analysis of gunshots recorded on a DPD dictabelt was presented by engineers from Bolt, Beraneck and Newman. The study indicated that there was a 96% probability that at least 6 sharp impulses occurred at 12:30 – the exact time of the shots. At least three were from in front of the limo at the Grassy Knoll. The evidence forced the HSCA to conclude that the assassination was “probably” a conspiracy. It was a “limited hangout”. There has been no follow-up investigation.

http://themysteriesofdealeyplaza.blogspot.com/2011/01/let-there-be-sound-acoustics-evidence.html

Sounds of Silence

“The conclusion of four separate shots coincides with 4 impacts visible in the Z-film. The acoustic impulses were retested in a 2001 investigation (‘Echo Correlation Analysis and the Acoustic Evidence in the Kennedy Assassination Revisited’) by D.B. Thomas, published in the Journal Science and Justice, Vol. 41, p. 21. The impulses are shown below, with the four highest amplitude peaks associated with rifle muzzle blasts.

The hypergeometric p-function was used for differing weighting factor distribution sets, H{M..N, n, i} to assess significance or likelihood of occurrence. It’s based on the no. of echo ‘windows’ M, with each spanning 190msec (total time) at 2msec width per window and n for assigned impulses in the evidence pattern, with ‘i’ the “coincident impulses” or those matching the original (11/22/63)evidence and the test result. The question was whether a succession of first impulses of given amplitude could be manifesting a signal or was merely random noise.

Thomas found that for a given configuration for 2 motorcycles at designated locations, 1 for (GK) shooter location and one for alignment of muzzle blasts with one pair of echoes, the p -value is 0.000012 or about 1 in 100,000 against the null hypothesis, i.e. that the impulses were from random noise. An alternative way to put this is that the odds are 100,000 to 1 in favor of the impulses comprising actual rifle shots.”

In 1965, assassination researcher and author David Lifton (“Best Evidence”) analyzed the famous Mary Moorman photo. He discovered a figure holding an object behind the fence on the Grassy Knoll. The image was the fifth, final and clearest image of the subject identified in the photo. Later, the figure was identified as a man by 10 independent photographic experts. Each signed statements to that effect and none were told that the photo was taken in Dealey Plaza. http://spot.acorn.net/jfkplace/09/Kelin33/no_5.html

The corporate media never discusses the HSCA conclusion of a probable conspiracy based on overwhelming eyewitness testimony and scientific acoustic evidence of shooters at the Grassy Knoll. Warren Commission apologists are relentless in their shameless promotion of the ridiculous, discredited Lone Nut Gunman and Single Bullet Theory.

Posted by on February 5, 2014 in JFK

### 19 responses to “JFK Assassination: Mathematical Proof of a Grassy Knoll Shooter”

1. February 6, 2014 at 10:18 am

The above is a bit like saying:

We asked 100 test subjects to pick a number in the range from 1 to 3, and 26 of them picked ‘1’, 42 picked ‘2’, and 30 picked ‘3’.
With a 1/3 probability of being right, the probability of being wrong is 2/3. Let’s calculate the probability that all the test subject who picked the number 2 were wrong. PN(2) = (2/3)^42 = 4.019E-8. Therefore, the probability that the right number is 2 is:
PS(2) = 1 – PN(2) = 0.99999995981.

Does this really prove the right number is 2? Of course not! The same (highly unscientific) methodology would give you PN(1) = 2.640E-5, and PN(3) = 5.215E-6, so the sum of the probabilities PS(1), PS(2) and PS(3) would be very close to 3. What does that tell you?
You can’t just sample the test subjects who give you the answer you like and ask: “how can they all be wrong?!”
There are other problems with your methodology, of course, but this will do for now.

• February 6, 2014 at 2:54 pm

That is an invalid “analogy” since it involves guessing a number with no other information. This is NOT the same as HEARING shots, SEEING smoke or seeing SHOOTERS at the Grassy Knoll.

In other words, what is important is the EVIDENCE. Your example is pure chance, like guessing a coin flip. It is totally bogus. The witnesses were not flipping a coin. They were relating their experience.

2. February 6, 2014 at 4:52 pm

Well, the cherry-picking was the same, but perhaps my example was slightly more honest, because I didn’t presume to know the “right” answer, and I didn’t pull the probability of being right/wrong out of thin air.

Here’s a little home assignment for you:

You claim it’s highly unlikely that 51 witnesses could all be wrong, and that the probability of a GK shooter is as high 0.999999999999139.

a) What is the probability, using your numbers and methodology, that there was a TSBD shooter?

b) How do the probabilities add up?

c) How do you account for the fact that only a small minority of the witnesses reported shots coming from more than one direction?

3. February 7, 2014 at 10:16 am

Mark, I must be blunt. Your comments confirm that you are clueless.

a) You figure out the probability of a TSBD shooter. I will tell you right now that it is virtually 100%, just like the probability of a Grassy Knoll shooter.

b) You ask how do the probabilities add up? That question immediately exposes you as having no knowledge of probability theory. I will try to educate you. The shots from the Grassy Knoll and TSBD are NOT MUTUALLY EXCLUSIVE events. The probability is virtually 100% for each. You don’t add them up. Now I know where you are heading with the question, so I will save you the trouble. Don’t tell me that because they add to more than 100% that the analysis is invalid. No, your understanding of probability is ZERO. Your statement confirms it.

c) How do I account for the fact that some witnesses heard shots coming from two directions? That is another ridiculous question which betrays your sheer ignorance. The answer is: Yes, they heard it. So what? They are correct. Shots came from two directions.

A final request. What is your math background? Did you ever take a probability course? It’s legitimate to ask questions. I welcome them. But you are being very presumptive to assert that my analysis is invalid. You see, I have three degrees in applied mathematics. How many do you have?

If you took a probability course, you probably failed. I say this based on the ignorant statements that you have made. But I bet you never took the course. I view you now as one very arrogant, ignorant poster. That’s a lethal combination.It’s one or the other. Whether or not you studied probability, you are exposed.

Do not reply unless you are willing to state your math background. If you reply with more blather maintaining your position, I will have no choice but to delete your comment as one who is unwilling to learn – and does not appreciate the essence of a factual math analysis.

4. February 7, 2014 at 4:47 pm

no dog in this fight but Mark said “few witnesses” reported shots from more than one direction

• February 8, 2014 at 4:23 pm

No, Ted, he said: c) How do you account for the fact that only a small minority of the witnesses reported shots coming from more than one direction?

And I answered him. So what? That is totally irrelevant. Yes, there were shots from both directions. I never said they were JUST from the Knoll. They are not mutually exclusive. I am calculating the probability that ALL witnesses who said shots were from the Grassy Knoll were MISTAKEN. VERY SIMPLE. I said nothing about the shots from the TSBD. THIS IS ABOUT THE GRASSY KNOLL…

5. February 9, 2014 at 7:20 pm

Just to get it out of the way: I did take (and pass) a course in Statistics and Probability Theory when attending University of Copenhagen many years ago. Perhaps more importantly, I have a degree in common sense and logic from the university of life.

It’s true that shots from the GK and shots from the TSBD aren’t mutually exclusive propositions, although your own data suggests that we’re more likely dealing with different perceptions of shots from a single direction (otherwise kindly explain why it appears that none of your witnesses said shots came from more than one direction). You’ve admitted that we can use your method to “prove” shots from the TSBD, as well as from the GK, so aren’t you (in a sense) saying that none of the witnesses got it right? It should bother you (as a supporter of the HSCA’s “acoustical evidence” findings) that your 51 “GK” witnesses either didn’t hear the 3 shots from the TSBD or mistakenly thought they came from the GK.

Well, you’ve made it clear that you don’t want to talk about shots from the TSBD, and I can understand why, but your method can even be used to “prove” that no shots were fired from the GK (I hope we can agree that shots coming from the GK and shots not coming from the GK are mutually exclusive). For the sake of clarity, I’m going to skip the undecided vote this time:

(Charnin mode ON)

Of 83 witnesses, 51 said shots came from the GK area, 32 from the TSBD.

Given the 32/83 probability of a witness being correct in stating that shots didn’t come from the GK, then the probability is 51/83 that the witness was mistaken.

Therefore, the joint probability (PN) that ALL 32 witnesses were mistaken and that shots came from the GK is:
PN = (51/83)^32 = 0.0000001705
Therefore, the probability that no shots came from the GK is:
PS = 1-PN = 0.9999998295

(Charnin mode OFF)

I probably don’t have to tell you that PS(shots from GK) + PS(shots not from GK) becomes > 1.

Frankly, your method doesn’t work.

6. February 9, 2014 at 8:28 pm

You cannot read or purposefully misquoted what I said. You claimed I said this:

(Charnin mode ON)
Of 83 witnesses, 51 said shots came from the GK area, 32 from the TSBD.

Given the 32/83 probability of a witness being correct in stating that shots didn’t come from the GK, then the probability is 51/83 that the witness was mistaken.

Therefore, the joint probability (PN) that ALL 32 witnesses were mistaken and that shots came from the GK is:
PN = (51/83)^32 = 0.0000001705

Therefore, the probability that no shots came from the GK is:
PS = 1-PN = 0.9999998295
(Charnin mode OFF)

No, this is what I said:
If we exclude the 38 witnesses who had no opinion as to the origin of the shots, then 51 of 83 (62%) said shots came from the Grassy Knoll and there is a 38% probability that a given witness would be mistaken. The probability that ALL 51 witnesses would be mistaken is much lower:
PA = 0.38^51 = 3.71E-22
PA= 1 in 2,697,966,622,402,536,000,000 or 1 in 2.7 billion trillion!

Mark, you are VERY confused…

1) OK, you took the course. Congratulations. But you need a refresher.
2) You see the light. You now agree that the differing observations of witnesses as to the origin of the shots are not mutually exclusive.
3) You miss the fact that ALL witnesses got it right, whether they said Grassy Knoll or TSBD. The acoustic evidence presented at HSCA proves it.
4) Those who offered no opinion can be excluded since shots were fired.
5) The fact that 51 witnesses heard shots or saw gunsmoke at the Knoll means the probability of any one being correct (based on the statistical data) is 0.62.
6) There are just TWO possibilities. Each witness is either correct or incorrect.
7) The probability a Grassy Knoll witness is mistaken is (1-0.62) = 0.38 (based on the 83 who offered an opinion).
8) The statistical data indicates a virtual 100% probability that shots were fired from the Knoll. Acoustic evidence presented at HSCA proved it. So does the Badgeman photo.
9) The statistical data indicates a virtual 100% probability that shots were fired from the TSBD. Acoustic evidence presented at HSCA proved it.
10) You have backtracked from the logical errors you made in your first two replies, but you still don’t get it.

11) You fail to appreciate that the point of this exercise was to prove that there was at least one shooter at the Grassy Knoll. The best available evidence are the observations of the witnesses. The odds are astronomical that there was a shooter otherwise EVERY single witness must have been mistaken. That is proof beyond any doubt. And that is the bottom line, isn’t it?

12) What does the acoustic and photographic evidence tell us about the origin of the shots? Does it confirm the mathematical probabilities based on witness observations?

13) Based on witness observations, what would be your best estimate that they would ALL be mistaken?

7. February 11, 2014 at 10:45 pm

We know what the probability of a listener accurately locating the source of a shot.
We know that from the BBN listener test performed at the time of the test shots. The results of that test were that the two observers correctly located the origin of the shot 82% of the time.

However, that number, as large as it is, is not really that important because we know that at least 33 people identified a shot originated from the grassy knoll. That is the real problem for the LN side on this issue.

Even if you assign a probability of only 10% that a listener would correctly identify the location of a shot (or 90% probability they got it wrong) it still means that the probability that all 33 people got it wrong is .9 to the 33rd power = .03 = 3%. So even if there is a 90% chance that each observer would get it wrong the probabilty that all 33 got it wrong is only 3%, meaning there is a 97% chance there was a shot from the Knoll.

While some may say that the 82% probability of getting it correct is too high, 10% of them getting it wrong is too low.

http://www.aarclibrary.org/publib/jfk/hsca/reportvols/vol8/pdf/HSCA_Vol8_AS_3_Earwitness.pdf

8. February 16, 2014 at 10:40 am

Apologies for not getting back to you sooner.

Your p = 0.42 (or 0.62 if non-GK and non-TSBD witnesses are disregarded) is not the probability of a witness being “right”. It’s rather the probability of a randomly selected witness (within the defined pool of witnesses) being a GK witness. We can view that as a positive result, if you like, but it is independent on whether a witness is actually “right” or “wrong”.

Likewise, your PN = (1-p)^51 is the probability of 51 out of 51 random selections being negatives.

Likewise, your PS = 1-PN is the cumulative probability of 0, 1, …, 50 out of 51 random selections being negatives.

Let’s face it. Your “51 flies (sorry: witnesses) can’t be wrong” argument is a logical fallacy and totally unscientific (attempts to enhance it with bogus math notwithstanding).

• February 16, 2014 at 10:58 am

PS: Sampling biases aside, we can probably all agree that the perception that shot(s) came from the GK area is well represented among the DP witnesses, but the idea that probability theory can magically turn “well represented” into “almost certainly correct” strikes me as very peculiar indeed. Especially for someone who claims to have three degrees in applied mathematics.

• February 16, 2014 at 3:09 pm

Yes, three degrees. Not a claim. It’s a fact.

You continue to have your head buried in the sand.

What is the mathematical definition of proof beyond a reasonable doubt in a court of law ? It’s defined as a 99% probability of guilt (or less than a .01 probability (E-02) of innocence). We are way beyond that.

There is less than 1 in a trillion trillion trillion probability (E-43) of 89 witnesses ALL being MISTAKEN in claiming that shots were fired from the knoll. But you cannot understand that.

E-43 is the probability that you know what you are talking about.

9. February 16, 2014 at 1:25 pm

As I stated in an earlier post, we know the results from an actual hearing test that was conducted at the time of the test shots.

Two observers were asked to determine the source of a shot. Combined they were able to correctly identify the source of the shots 82% of the time when the shots were from either the Grassy Knoll or the TSBD.

http://www.aarclibrary.org/publib/jfk/hsca/reportvols/vol8/pdf/HSCA_Vol8_AS_3_Earwitness.pdf

We also know from the test shot acoustical recordings that there were no significant echos from the Grassy Knoll area.

The following image identifes the location of echos for one of the test shots from the TSBD. There were no significant echos from the Knoll area.

It should not be surprising that there is no problem discriminating the origin of a shot between the knoll and the TSBD.

However, it would be a problem for listeners if they were trying to discriminated between a shot from the TSBD and the DALTEX or Criminal Records Building.

• February 16, 2014 at 2:41 pm

Gknoll,

Thanks for that. It is perfectly stated, logical and scientific. The analysis is based on actual observations, just like those of nearly 100 witnesses who heard shots coming from the Grassy Knoll area.

10. February 16, 2014 at 1:43 pm

Mark,

You won’t give up, will you? You must have given much thought to your comment. But it still won’t fly. You see, it’s not just math; it’s common sense. Do you really believe that ALL witnesses who said there were shooters at the knoll would be mistaken?

UPDATE: I have new info for you. Of 176 witnesses, 89 said shots came from the front, 44 from the rear, 3 from both directions, 4 said elsewhere.

OK, let’s start all over again.

Review:
A given percentage P of witnesses determined that shots came from the Knoll. We can assume the probability P of ANY ONE being correct as a BEST ESTIMATE BASED ON ACTUAL WITNESS OBSERVATIONS. P IS A FUNCTION OF THE BEST OBSERVABLE SENSORY DATA: WHAT THEIR EYES, EARS AND NOSE IS TELLING THEM.

The BEST ESTIMATE of the probability that a given witness is MISTAKEN is 1-P.

EXAMPLE:
Consider a baseball player with a .300 lifetime batting average. His BA is the PROBABILITY OF GETTING A BASE HIT IN HIS NEXT AT BAT. IT IS BASED ON OBSERVATIONAL PERFORMANCE.

The probability that the player would FAIL to get a base hit in 80 consecutive at bats is
.7^80 = 4.05E-13 = 0.000000000000405 or 1 in 2,466,929,843,521

Has a lifetime .300 hitter EVER failed to get a base hit in 80 consecutive at bats? Look it up.

Oh, I forgot to ask:
DO YOU BELIEVE THERE WAS A SHOOTER IN THE KNOLL AREA?
OR DO YOU BELIEVE THAT ALL THE SHOTS WERE FIRED FROM THE TSBD?
IN EITHER CASE, WHAT IS THE BASIS FOR YOUR BELIEF?

• February 16, 2014 at 3:37 pm

Are you for real? A guy with a .300 BA probably won’t MISS base 80 times in a row, but your claim is that he’ll HIT base 80 times in a row (which is even more unlikely).

I would, of course, expect several hits and several misses, and likely more misses than hits. If we assume a binomial distribution applies (with n=80 and p=0.3), then we have, for example, P(18≤X≤38) = 0.95. Assuming our batter is reasonably consistent and conditions are within the norm, I’d feel safe about putting my money on somewhere between 18 and 38 hits out of 80.

• February 16, 2014 at 4:47 pm

Probably won’t get a hit 80 times in a row? It’s not probably. It never happened. And are you so brain-dead that you are claiming that I said he will get a hit 80 times in a row?

YOU ARE ONE VERY CONFUSED DUDE.
YOU ARE EMBARRASSING ALL RATIONAL READERS HERE.

You did not answer my questions.
DO YOU BELIEVE THERE WAS A SHOOTER IN THE KNOLL AREA?
OR DO YOU BELIEVE THAT ALL THE SHOTS WERE FIRED FROM THE TSBD?
IN EITHER CASE, WHAT IS THE BASIS FOR YOUR BELIEF?

Now it’s too late. Mark, that’s it. We are done here. It’s time for you to go. Don’t bother replying. I will not post it.

But thanks for giving the readers an education on how not to analyze a problem in simple probability and statistics.

• February 21, 2014 at 11:39 pm

Mr. Charnin,

You seem to have ignored a key point of Mr. Ulrik’s. Namely, you claim that the fact that 42% of witnesses believe shots came from the depository means that there is a 42% chance that each individual is correct. Let me give an example: there was a poll that showed that 28% of Americans think Abraham Lincoln was a Democrat. But there’s not a 28% that each of them was correct; there’s a 0% chance. And since Pew surveyed 1,000 people, your logic says that there’s a .72^280 = 1e-40 chance that they’re all wrong. Or we can flip your argument on its head. 42% of people heard those shots. Thus, they have a 42% chance of each being right. Thus, there’s a .42^51 = 6e-20 chance that they’re all correct! In fact, given any observation held by a small minority, you can argue, given a large enough sample, that they MUST be right. Imagine this: 50,000 people at an NFL game. 3% (1,500) think they saw a flying saucer, 97% disagree. Therefore, by your logic, there’s a .97^1500 = 1.4e-20 chance that those 3% are wrong. Even if just 1% “saw” it, there’s a .99^500 = .0065 chance. Even you have to see the flaw here. I am no expert on the JFK assassination, and I don’t believe that I have either the necessary knowledge of the evidence or the expertise to say that there weren’t shots from the Grassy Knoll. Perhaps there were. But this “statistical analysis” does not bolster your cause.

• March 3, 2014 at 4:53 pm

Your Lincoln polling example is ridiculous. It was not an EVENT. It was a poll. Dealey Plaza witnesses were asked about an actual event. It was not a poll. It was an event that they observed. They were asked what they observed. So much for honest Abe.

You wrote: “In fact, given any observation held by a small minority, you can argue, given a large enough sample, that they MUST be right. Imagine this: 50,000 people at an NFL game. 3% (1,500) think they saw a flying saucer, 97% disagree. Therefore, by your logic, there’s a .97^1500 = 1.4e-20 chance that those 3% are wrong. Even if just 1% “saw” it, there’s a .99^500 = .0065 chance”.

First of all, you are implying that UFOs do not exist. UFOs are seen virtually every day. So your premise is wrong from the start. Do you know how many were seen in the famous 1994 Phoenix, AZ incident? Were the witnesses ALL delusional?

If 1,500 of 50,000 people at a football game independently claimed they witnessed a UFO, there can be no question that they saw SOMETHING. They cannot ALL be delusional. So what are the odds they were ALL mistaken and saw nothing? ZERO. Do you disagree with that?

This is about 120 Dealey Plaza witness INDEPENDENT observations. What makes you think that they were ALL mistaken? What makes you think that ALL the shots came from the TSBD? See, I can play your game, too.