Probability of logical disjunctions¶

All other rules of probability theory can be derived from the two basic rules. So take two logical propositions $$A$$ and $$B$$. The logical disjunction of $$A$$ and $$B$$ is the logical proposition ($$A$$ or $$B$$). In our notations we write $$A + B$$ for ($$A$$ or $$B$$) We can deduce from the basic rules of probability that:

$p(A + B|I) = p(A|I) + p(B|I) - p(AB|I).$

In words, this says

The probability of A or B is the probability that A is True plus that probability that B is true minus the probability that both A and B are True.

Fig. 5 Venn diagram showing the information $$I$$, and the logical propositions $$A$$ and $$B$$.

This is very easy to understand intuitively by looking at the Venn diagram. Fig. 3. The probability $$p(A+B|I)$$ is the area of the union of A with B (normalized by I). This area is indeed the area of A (normalized by I) plus the area of B (normalized by I) minus the area of A and B (normalized by I) which was double-counted.

Let’s see a formal proof of this:

$\begin{split} \begin{split} p(A+B|I) &=& 1 - p(\neg (A+B)|I)\\ &=& 1 - p(\neg A, \neg B|I)\;\text{(obvious rule)}\\ &=& 1 - p(\neg A|\neg B, I)p(\neg B|I)\;\text{(product rule)}\\ &=& 1 - \left[1 - p(A|\neg B, I)\right]p(\neg B|I)\;\text{(obvious rule)}\\ &=& 1 - p(\neg B|I) + p(A|\neg B, I)p(\neg B|I)\\ &=& 1 - p(\neg B|I) + p(A\neg B|I)\;\text{(product rule)}\\ &=& 1 - p(\neg B|I) + p(\neg B|A,I) p(A|I)\;\text{(product rule)}\\ &=& 1 - p(\neg B|I) + \left[1 - p(B|A,I)\right]p(A|I)\;\text{(obvious rule)}\\ &=& 1 - p(\neg B|I) + p(A|I) - p(B|A,I)p(A|I)\\ &=& 1 - \left[1 - p(B|I)\right] + p(A|I) - p(B|A,I)p(A|I)\;(\text{obvious rule})\\ &=& p(A|I) + p(B|I) - p(B|A,I)p(A|I)\\ &=& p(A|I) + p(B|I) - p(AB|I)\;\text{(product rule)}. \end{split} \end{split}$