Peace Education in Math Classroom
TOYOSHIMA Kouichi
Faculty of Science and Engineering, University of Saga
1 Honjo-machi, Saga, 840-8502 Japan
E-mail: toyo@cc.saga-u.ac.jp

abstract
Probabilistic consideration shows that the chances of being
aggressed and of being an aggressor are mathematically equal.
This simple "theorem" may help people to assess its own army asa potential invader as well as an alleged "defence" force.
Armed forces are kept by alleging, without exception, that its purpose is to"defend" its own country and never for invasion to another country. If one
argues for the abolition of force, he will be challenged by the question like
"What if another country invades us?" This is also the case for the advocates ofthe Article 9 of the Japanese Constitution. Although this is a natural questionand has to be answered anyhow, one has to remind that the following question
should equally be posed: "What if our army invades another countries?"
However, we hardly hear this second question, not only in Japan. This may bebased on our misconception that the former case is mathematically moreprobable, because the number of potential invader to his/her country is
numerous but his/her country which may invade is single. But this is simply
false. So it may be essential to 'reset' this popular fallacy and to remindourselves that these possibilities are mathematically equal.
The following is an example of "theory" which may serve as a remedy for thismisconception. It is not a realistic discussion but only a simple algebra
indicating the starting point for, if you want, more sophisticated probabilistic
theories. Nevertheless I believe it will help one to remind that these twoquestions, namely about the risk of being invaded and of becoming an invader,should be asked equally, equally frequently and equally seriously.
Suppose we have n+1 states whose probabilities of invasion to another states
are all equal. Let p be this probability of invasion by some state to any otherstates in a certain period of time T. The quantity p is also an expected value ofthe frequency of invasion by a state in this period. Then the expected frequency
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of invasion by a state to one of other states is p/n. We assume no correlation
among the behavior of different countries, e.g. alliance, exists.
Let us calculate the expected frequency of invasion to some state by any otherstates. The probability P(k) of being invaded simultaneously during the period Tby k states isp k p  n−k
P( k ) =1−
 n n

and its number of cases is n C k .
Then the total expectation of frequency of invasion by all of other states isobtained by summing up all possible values of k weighted by n C k P(k):nnp kp n−kf = ∑ k n C k P( k ) = ∑ k n C k    1 −  n nk =1

k =1kn−kn!
 p  1 − p  n n
k =1 ( n − k ) ! ( k − 1) !
n=∑

Replacing k with r +1 , this expression becomesn!
p  r +1 
p  n − r −1f =∑1− n n
r = 0 ( n − r − 1) ! (r ) !
n −1

p=n•
n=n•

rn −1− r(n − 1) !
 p  1 − p ∑ n n
r = 0 ( n − r − 1) ! (r ) !
n −1p pp n −1+1−  = pnn nNamely, f is simply equal to p.
This result is quite an anticipated one because "defensive war" and "aggression"
are only the different names for the same event, the war. This simple
mathematical inference could be used as a teaching stuff in high school or incollege math classrooms to promote the idea for peace and disarmament.

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