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Terrorist attack |
President Bush has stated his support for a significant increase in the
number of air marshals and he has asked for $500 million in appropriations
for that program and other air safety measures. The number of marshals the
President requested is classified to not reveal defense capability to
terrorists. Unfortunately, keeping the number of marshals secret will not
keep terrorists from concluding correctly that the announced air marshal
program will provide little security.
In the NewsMax article Armed Citizens the Best Defense Against Terrorists,
Sept. 28, 2001, Professor John Lott is quoted to say that 35,000 officers will
be required to place one air marshal on each of the daily flights. But the
amount of money requested by President Bush is a small fraction of that needed
to fund 35,000 marshals.
To examine security implications of this limited funding, we analyze the
impact on security of two levels within the $500 million budget constraints.
The first analysis assumes a level of 3,000 air marshals or approximately a
hundred fold increase over current levels. At an average cost for salary
and benefits of $50,000 per marshal, 3,000 marshals will cost $150 million
plus some administration and overhead (say another $50 million). That
amount is approximately two-thirds of the cost of three Boeing 757's (the
type of airplane used in the Sept. 11 attack) and two-fifths of 1% of the
cost of clean-up and replacing the World Trade Towers in New York. So the
cost looks reasonable and one hundred fold increase in marshals looks like a
lot of security doesn't it?
Well, maybe not so much security once we consider how terrorists might view
this program. A terrorist planner could perform a parametric study just as
we do here. He will know that adding attack groups increases the likelihood
of finding planes without marshals. Our goal is to see how many attack
groups might be needed and how that number is driven by the number of air
marshals employed.
We assume for this analysis that a terrorist planner wants to gain control
of three aircraft just like the Sept. 11 attack where three attempts
succeeded and the fourth was defeated by unarmed ordinary citizens. He
might assume that a single marshal on a aircraft could defeat the attack on
that aircraft. Whether or not true, planning for 100% marshal effectiveness
will give the terrorist one view of the attack results. Assume that the
terrorist planner wants a 95% chance that his groups will be able to gain
control of three planes and that the only barrier is that provided by air
marshals (i.e. without a marshal on the aircraft the terrorist group will
gain control). The planner needs more than three attack groups to account
for some being defeated by marshals. Statistics can tell us how many more
will be needed.
The terrorist planner is looking for the number of attack groups, say N,
that results in at least three groups being unopposed by marshals. That is,
the number of marshals encountered should not be more than N-3. We assume
marshals are randomly assigned to the flights with a equal probability for
all flights (and the terrorists select aircraft randomly to attack). The
probability of a marshal being on any particular flight we denote as P.
If the number of marshals is 3000, the probability of a marshal being on a
particular aircraft is
P = 3000/35000 = .0857
Let Q denote the probability that no marshal is on a particular flight.
Q = 1 - P = .9143
The probability of exactly M marshals on the N flights attacked is
P(M|N) = {N|M}Q**(N-M)*P**(M)
where "**" denotes exponent (power), {N|M} = N!/[M!*(N-M)!] and ! denotes
factorial.
With N groups (N>3) attacking, the probability that there will be no more
than N-3 marshals on the aircraft (hence three or more are uncovered) is
given by
PS(N) = P(0|N) + P(1|N) + ... + P(N-3|N)
i.e. (Prob. of zero marshals) plus (Prob. of one marshal) plus ... etc.
PS(N) is the probability that the terrorists succeed in gaining control of 3
aircraft by attacking N. The following table shows the probability of
terrorist success with N groups attacking (N = 4, 5 and 6) if there are 3,000
air marshals:
N PS(N)
4 .961
5 .994
6 .999
An attack similar to Sept. 11 (with four attack groups) could be conducted
with better than a 95% chance of success of gaining control of three or more
aircraft if the only protection is the 3,000 air marshals. This hundred-fold
increase in air marshals will not have much chance (alone) of defeating a
Sept. 11 attack. More specifically, the statistics tell us that employing
3,000 air marshals have no deterrent value against a Sept. 11 style attack.
Note that we are not implying that individual air marshals would not be
effective. To the contrary, we assume they are 100% effective in stopping
an attack when present. The problem is that the number of marshals must be
sufficient to cover the daily flights. Without that number of marshals
there are significant holes in security that terrorists can exploit.
If the full $500 million set aside by President Bush were to be used for air
marshals and none for strengthening the aircraft cockpit or other security
measures the money requested by President Bush could provide 7,500 air
marshals. What would that number of marshals do to prevent a Sept. 11 type
attack?
The probabilities P and Q above change, but the methodology does not. In
this new case the probability of finding a marshal on a particular plane
would be
P = 7500/35000 = .214
and the probability no marshal will be on a particular plane is
Q = 1 - P = .786
Calculating terrorist probability of success for four, five, and six attack
groups as before yield for 7,500 air marshals the table:
N PS(N)
4 .797
5 .931
6 .978
So, with 7,500 air marshals the terrorist group would have to launch six
attacks before they would have a 95% level of certainty that at least three
would succeed. Clearly, even without knowing the "classified number" of
marshals, a terrorist planner can estimate how marshals will degrade an
attack. The principal value in the 7,500-level vice 3,000-level of air
marshals is to force the terrorist organization to increase by 50% the
number of attack groups to accomplish the same level of success. The cost
for the 7,500-level of air marshals is 250% of the 3,000-level.
Terrorists might think an attack to be a success even if only one of the
aircraft hit its target. The probability that a terrorist attack on four
aircraft would find at least one uncovered by a marshal at the 7,500-level
is 0.998 and that two will be uncovered is 0.967. To insure that even one
attack cannot succeed will require nearly comprehensive coverage of aircraft
by marshals. As many as 30,000 marshals only have a .54 probability (one
chance in two approximately) of being on all four aircraft for a Sept. 11
style attack.
It seems clear that President Bush's air marshal program will not have any
serious deterrent or damage prevention value against another terrorist
attack similar to the one made on Sept. 11 and that security rests primarily
with the other measures being taken.
Since he has issued an order permitting two air force generals to decide when
to shoot down civilian airliners, we conclude that President Bush has little
confidence in his other security measures. While that shoot-down step is
likely to mitigate the actual damage terrorists might do using the aircraft
as bombs, it will not do much for those finding themselves on an aircraft
attacked.
We question the moral justification of a government program which permits
the United States Armed Forces to kill innocent civilians while denying them
the ability to save their own lives when attacked. If armed opposition to
terrorists is an important component to preventing terrorist from gaining
control of an aircraft, the Federal Government should provide that armed
guard immediately one way or another.
Philip F. Lee
12921 Two Farm Dr.
Silver Spring, MD 20904
301-622-0446 Home
703-418-8225 Work
Philip F. Lee has a PhD in Mathematics from Georgia Institute of Technology, 1970.
He has more than 28 years experience in the application of statistics to defense
and industrial applications.
Added on 10/2/01 by Phil Lee.