The tragedy in Oregon has reignited the gun debate. Gun control advocates argue that fewer guns mean fewer deaths while gun supporters argue that if citizens were armed then shooters could be stopped through vigilante action. These arguments can be quantified in a simple model of the probability of gun death, 
:
where
is the probability of having a gun,
is the probability of being a criminal or mentally unstable enough to become a shooter,
is the probability of effective vigilante action, and
is the probability of accidental death or suicide. The probability of being killed by a gun is given by the probability of someone having a gun times the probability that they are unstable enough to use it. This is reduced by the probability of a potential victim having a gun times the probability of acting effectively to stop the shooter. Finally, there is also a probability of dying through an accident.
The first derivative of
with respect to
is
and the second derivative is negative. Thus, the minimum of
cannot be in the interior
and must be at the boundary. Given that
when
and
when
, the absolute minimum is found when no one has a gun. Even if vigilante action was 100% effective, there would still be gun deaths due to accidents. Now, some would argue that zero guns is not possible so we can examine if it is better to have fewer guns or more guns.
is maximal at
. Thus, unless
is greater than one half then even in the absence of accidents there is no situation where increasing the number of guns makes us safer. The bottom line is that if we want to reduce gun deaths we should either reduce the number of guns or make sure everyone is armed and has military training.
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