# How to express disjunction in Probability Theory?

Joint probability is usually expressed with a comma:
$P(A=a, B=b)$ meaning: The probability of random variable A having value a and random variable B having value b

But what would be the notation of the disjunction? The probability of random variable A having value a or random variable B having value b? $P(A=a ? B=b) = P(A=a) + P(B=b) - P(A=a, B=b)$

I could use the logical 'or' $P(A=a \vee B=b)$ but then shouldnt i also use the logical 'and'?

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Demanding someone to use the "and" is the same as demanding someone to write $2 \times x$ instead of $2x$. –  André Caldas Nov 7 '11 at 16:18

## 1 Answer

Be aware that usually in probability theory $\wedge$ (logical and) denotes minimum: $$a\wedge b = \min\{a,b\}$$ so for the conjuction and disjunction one usually uses $$\mathsf P(\{A = a\}\text{ and }\{B = b\}) = \mathsf P(\{A = a\}\cap \{B = b\})$$ $$\mathsf P(\{A = a\}\text{ or }\{B = b\}) = \mathsf P(\{A = a\}\cup \{B = b\})$$ thinking about events $\{A = a\}$ and $\{B = b\}$ as sets due to measure-theoretical approach to probability theory. On the other hand, joint events appeared more important for problems of probability theory so that may be the reason why probability of joint events has a shorthand.

For the disjunction I never seen any other shorthand but $\cup$.

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Thanks for the answer. This came up because one of the axioms of probability theory involves disjunction (you know the one: P(A U B) = P(A) + P(B) for A and B mutually exclusive). Every book on belief networks starts with the axioms, either informally of in terms of sets of events from the event space. In the latter case probability theory is extended to beliefs or variable assignments after the axioms are introduced. The way i see it, if i'm writing about beliefs or variable assigents, why not define the axioms in those terms. –  Ivana Nov 10 '11 at 16:14
@Ivana: that may certainly differ for several fields of math, e.g. I sometimes see $\int$ to denote not an integral but rather a time which system spend in some state. I just can tell you that I've never seen such notation in book on classical PT and that $\wedge$ and $vee$ there usually denote $\min$ and $\max$, while $\cap$ and $\cup$ refers to the set operations. –  Ilya Nov 10 '11 at 16:27