Addition Theorem - Kahouteiri

Japanese: 加法定理 - かほうていり
Addition Theorem - Kahouteiri

There are many types of addition theorems in mathematics, but here we will introduce the following two as the most common ones.

(1) The addition theorem of trigonometric functions is the following formula.


(2) Addition theorem in probability Suppose there are k events E1 , E2 , ..., Ek , and no two of them occur simultaneously. In other words, E1 , ..., Ek are mutually exclusive events. In this case, the probability p that at least one of E1 , ..., Ek occurs is equal to the sum of the probabilities p( Ei ) of the occurrence of each event Ei.

p=p(E 1 )+p(E 2 )+……+p(E k )
This is the addition theorem in probability. In classical probability theory, this theorem is proven as follows: "Suppose there are a total of N possible cases, and each case is equally likely to occur. If there are n i possible cases corresponding to event E i , then p(E i )=n i /N. On the other hand, since no two of E 1 ,……,E k can occur simultaneously, the number of cases in which at least one of these k cases occurs is N=n 1 +……+n k . Therefore,

When constructing probability theory axiomatically, the following property is considered to be an axiom regarding probability: "When E1 and E2 are mutually exclusive events, the probability p( E1 ∪E2 ) that at least one of E1 and E2 occurs is equal to the sum of p( E1 ) and p( E2 )."

[Shigeru Furuya]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

数学において加法定理とよばれているものはいろいろあるが、ここではもっとも一般的なものとして次の二つをあげる。

(1)三角関数の加法定理 次の公式をいう。


(2)確率における加法定理 k個の事象E1,E2,……, Ekがあって、このうちのどの二つも同時におこることはないとする。すなわちE1,……,Ekが排反事象であるとする。このときE1,……, Ekのうちの少なくとも一つがおこるという確率pは、各事象Eiのおこる確率p(Ei)の和に等しい。

  p=p(E1)+p(E2)+……+p(Ek)
これが確率における加法定理である。この定理は古典的な確率論では次のようにして証明される。「全部でN通りの場合があって、どの場合がおこるのも同様に確からしいとする。事象Eiに対応する場合がni通りであるとするとp(Ei)=ni/Nである。一方、E1,……, Ekのうちのどの二つをとっても同時におこることはないから、これらk個のうちの少なくとも一つがおこる場合の数はN=n1+……+nkである。したがって

である」。確率論を公理的に構成するときは、「E1、E2が排反事象であるとき、E1、E2のうちの少なくも一方がおこる確率p(E1∪E2)は、p(E1)とp(E2)の和に等しい」という性質を確率に関する公理と考える。

[古屋 茂]

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