Mutually exclusive and exhaustive

A set of propo­si­tions is “mu­tu­ally ex­clu­sive and ex­haus­tive” when ex­actly one of the propo­si­tions must be true. For ex­am­ple, of the two propo­si­tions “The sky is blue” and “It is not the case that the sky is blue”, ex­actly one of those must be the case. There­fore, the prob­a­bil­ities of those propo­si­tions must sum to ex­actly 1.

If a set \(X\) of propo­si­tions is “mu­tu­ally ex­clu­sive”, this states that for ev­ery two dis­tinct propo­si­tions, the prob­a­bil­ity that both of them will be true si­mul­ta­neously is zero:

\(\forall i: \forall j: i \neq j \implies \mathbb{P}(X_i \wedge X_j) = 0.\)

This im­plies that for ev­ery two dis­tinct propo­si­tions, the prob­a­bil­ity of their union equals the sum of their prob­a­bil­ities:

\(\mathbb{P}(X_i \vee X_j) = \mathbb{P}(X_i) + \mathbb{P}(X_j) - \mathbb{P}(X_j \wedge X_j) = \mathbb{P}(X_i) + \mathbb{P}(X_j).\)

The “ex­haus­tivity” con­di­tion states that the union of all propo­si­tions in \(X,\) has prob­a­bil­ity \(1\) (the prob­a­bil­ity of at least one \(X_i\) hap­pen­ing is \(1\)):

\(\mathbb{P}(X_1 \vee X_2 \vee \dots \vee X_N) = 1.\)

There­fore mu­tual ex­clu­sivity and ex­haus­tivity im­ply that the prob­a­bil­ities of the propo­si­tions sum to 1:

\(\displaystyle \sum_i \mathbb{P}(X_i) = 1.\)

Parents:

  • Probability theory

    The logic of sci­ence; co­her­ence re­la­tions on quan­ti­ta­tive de­grees of be­lief.