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APPENDIX A 35
APPENDIX A
CRITICISM OF PROBABILITY CALCULATIONS
A-L. CRITICISM OF BRSW PROBABILITIES 0.88, 0.88, 0.50, AND 0.75
This criticism refers to the calculation by BRSW of probabilities of 0.88, 0.88, 0.50, and 0.75 for the
identification of 4 impulse patterns of the DPD tapes as representing four shots. The third pattern is associated
with the conjectured shot from the grassy knoll. These claimed probabilities are, at the very least, larger than the
BRSW reasoning should permit, if that line of reasoning were to be accepted.
In Appendix C, pages C1-C2, of the BRSW report2, it is calculated that the 432×4 correlation matches of
the 432 echo patterns (derived from the test shots) with the four impulse patterns (on the DPD tapes that were
suspected of being patterns from shots) should give an expected 13 false alarms. BRSW found 15 matches,
which is within reason. (See Section 5.3 and Figure 22 on pages 60–63 of that report). BRSW applied a 2×2
contingency table test based on the data in their figure 22 to decide that the matches are not randomly located
and consequently at least two of those matches must be real. (See page 66 of BRSW).
We shall comment on this conclusion later, but let us grant this conclusion for the time being. BRSW
observe that six of the matches are clearly false alarms. This leaves nine points of which at least two are true
matches. We quote BRSW on page 66.
“However, the expected number of false alarms to be found when testing four different impulse patterns is 13 (see
Appendix C), and only six have been found. Therefore, it is not unreasonable to expect that there are seven more,
although that would be the largest number possible since at least two of the remaining nine
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APPENDIX A 36
are probably detections. The best that can be safely assumed is that each of the nine remaining correlations is
equally likely to represent a detection or a false alarm.”1
The intended meaning of that last sentence is that each of the nine candidates has as large a probability of
being a true match as any other candidate. We would suggest that a conservative estimate of this probability
based on the BRSW reasoning would be 2/9. Instead, BRSW apparently misinterpreted their own words, reading
“equally likely” to mean that the probability of a false alarm for each detection is 1/2, an interpretation consistent
with the somewhat ambiguous language but not with the reasoning.
From their interpretation they proceed to calculate the probabilities 0.88=1−(1/2)3, 0.88=1–(1/2)3, 0.5=1–
1/2, and 0.75=1−(1/2)2 on p. 67, using the probability 0.5, the questionable assumption of independence and the
fact that of the remaining nine matches which are not obviously false alarms three correspond to the first
conjectured shot, three to the second, one to the third (grassy knoll) and two to the fourth (see Figure 22 of
BRSW report).
Using 2/9 instead of 1/2 would give probabilities 1−(7/9)3=0.53, 1−(7/9)3=0.53, 1−(7/9)=0.22, and l−(7/9)
2=0.40. These are considerably smaller than the BRSW probabilities.
One may argue that the above calculations are too conservative and that the probability 2/9 should be
replaced by a larger number, say 3/9 or 4/9. Certainly the use of 4/9 would be unduly “optimistic” since there are
at most four shots.
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APPENDIX A 37
A-2. CRITICISM OF BRSW CERTAINTY THAT MICROPHONE DETECTED SOUND OF
GUNFIRE (P. 64)
One must question the reasoning that led to the inference that at least two of the alarms were “true”. The
2×2 chi-square contingency table analysis used depends on the assumption of 15 independently located alarms.
Those alarms corresponded to microphone and rifle locations closely grouped in space. Hence the signals are
similar and one should expect that the high correlation coefficients for these alarms are highly dependent events.
An informal study of these locations and Figure 22 would suggest that these 15 alarms are effectively equivalent
to fewer, say about 7 or 8, independent points. In that case the significance of the layout is considerably reduced
and one may question the conclusion on page 64 of the BRSW report that “the motorcycle was moving through
Dealey Plaza and did, in fact, detect the sounds of gunfire.” (To be specific, a two by two table with entries
1,3,4,0 yields a significance value P=0.07, rather than the P<0.01 claimed by BRSW.)
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APPENDIX A 38
A-3. CRITICISM OF BRSW/WA PROBABILITY OF 0.95 FOR SHOT FROM GRASSY KNOLL
The BRSW/WA conclusion of a 95% probability of a shot or loud noise from the grassy knoll fails to be
convincing because (i) the use of subjective procedures which easily lead one to unconsciously biased reporting
and make it difficult to reproduce the results by independent observers, (ii) serious errors in statistical reasoning
which render the calculated probabilities meaningless, and (iii) the failure to apply the WA methodology to the
other suspected shots which leaves the method insufficently tested and calibrated.
We elaborate here on (ii). First the BRSW calculations on page 75, using the claim of coincidence of 10 out
of 12 predicted echoes with 10 of the 14 impulses on the DPD tape that exceeded a threshold, should conclude
that the probability of observing 10 or more coincidences is less than 0.053 under a Poisson randomness
hypothesis. It does not follow that the Poisson randomness hypothesis has probability less than 0.053. Such a
conclusion is no more valid than it would be to conclude that the dealer of a bridge hand who deals himself 3
aces on the first deal is dishonest with probability 0.96. This first inference would require the application of
Bayes Theorem using prior probabilities for all plausible hypotheses, and the probabilities of the observation of
10 matches under each of these plausible hypotheses. In the case of the bridge dealer, a probability assignment to
the hypothesis of honesty would depend on one's prior belief in the dealer's honesty and would also involve the
calculation of the probability of his dealing himself 3 aces on the first deal if he were dishonest.
Second, the calculation of 0.053, which is typically called the significance level or P value, should have
included an adjustment to allow for the fact that the hypothesis of a shot from the grassy knoll involved 7 “free”
parameters that were adjusted to maximize the number of coincidences. These parameters were time of shot,
position of shooter (2), initial position of motorcycle (2), velocity of motorcycle and velocity of sound combined
with tape recorder speed. The adjustment used by BRSW/WA is somewhat ad hoc, depends mainly on adjusting
three of these parameters
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APPENDIX A 39
(initial position of motorcycle and time of shot), and requires review. That adjustment consists approximately of
reducing the number of matches by one and multiplying the computed probability (3.13×10−4) of as good a
match by a factor of 180. The latter factor was based on the number of plausible initial positions of the
motorcycle that could have led to different results in matching predicted echoes against observed impulses. But
if they had chosen to consider different plausible values for the other parameters, their reasoning would have
produced a greater factor. For example, the 50-foot range of plausible shooter positions along the fence could
contribute an additional factor of 5 to 10, since movements of more than ±5 feet could change the relative
positions of the predicted echoes by substantially more than could be compensated for by readjusting the initial
position of the microphone (see page 29 of WA). A more traditional and possibly overconservative adjustment
would consist of subtracting one match for each free parameter. This adjustment would lead to a less significant
result (high P value).
Third, alternative hypotheses to the two primarily considered (gunshots or random locations of impulses
according to a Poisson process) should have been considered, such as non-white (non-Poisson) noise and static.
Such distributions could increase the likelihood of the BRSW and WA results having been obtained by chance.
Fourth, the calculation of 0.053 involved some further errors. On page 75, the BRSW calculations claim
that 12 of 22 predicted echoes were loud enough to exceed a threshold. (It seems that 22 should be 26 if Table 4
on p. 27 of WA is the source.) Then 10 of these 12 predicted echoes occurred within ±1 msec. of the occurrence
of 10 of 14 impulses on the DPD tape that were loud enough to exceed a threshold. Later there is reference to
two time intervals of 90 msec total duration, representing forty-five 2-msec. windows.
The time should be 180 msec representing 90 windows. In two cases a pair of impulses correspond to a
single window. (See Figure 7, page 28 of WA.) These are marked (19, 20) and (23, 24). For the BRSW
hypergeometric probability calculations to be appropriate, it is necessary to use
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APPENDIX A 40
non-overlapping windows and to count as coincidences the number of windows in which there are at least one
predicted echo and at least one observed impulse. Moreover, two of the predicted echoes appear in one window.
Thus 10 coincidences among 12 predicted echoes and 14 impulses out of 45 windows should be adjusted to 8
coincidences among 11 predicted echoes and 12 impulses out of 90 windows. If we now reduce the number of
concidences by 7 to make a conservative adjustment for the “free” parameters, we would have 1 coincidence
among 4 predicted echoes and 5 impulses out of 83 windows. The hypergeometric probability calculation would
then yield a significance level of P=0.223, which is not at all impressive in contrast to the claim that P=0.053.
However, this adjustment may be unduly conservative.
In summary,
(1) The BRSW/WA conclusion of a probability of 0.5 of a shot from the grassy knoll on the basis of the
BRSW analysis is invalid as is also the conclusion of a probability of 0.95 for such a shot on the
basis of the WA analysis.
(2) There are several inaccuracies.
(3) Except for the rather conservative analysis above, the data do tend to cast doubt on the hypothesis of
random impulse locations according to a Poisson process.
(4) Alternative hypotheses to a random Poissson process and a shot should have been examined as
possible explanations of the coincidences. These might invoke the nature of the bursts of noise
prevalent during the period under study and a consideration of other possible non-Poisson
distributions.
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