Let $S = \{1, 2, \dots, 10\}$ Let $A$ be the event that a number selected from $S$ is even. Let $B$ be the event that a number selected from $S$ is a multiple of $3$.
If any number is equally likely to be selected from $S$, the probability that a number selected from the set $S$ is neither even nor a multiple of $3$ is $\frac{3}{10}$. I write it as $P(\overline A \cap \overline B) = \frac{3}{10}$.
But I have some people writing it as $P(\neg A \text{ and } \neg B) = \frac{3}{10}$. I don't understand the reason for this notation. They seem to be using logical operators which apply to statements in propositional logic. But $A$ and $B$ are not statements. They are events, and events are sets. Shouldn't we stick strictly to set theory notation while discussing probability?
If we have to use logical operators, shouldn't we use $P(\neg A \land \neg B)$ instead of $P(\neg A \text{ and } \neg B)$?
Best Answer
If you have a sample space $\Omega$, then you can consider an event to be either a subset of $\Omega$, or a logical formula $\phi(x)$ parameterized by an element $x$ of $\Omega$ (here we might formally consider $x$ an $\Omega$-valued random variable, namely the identity $\Omega\to\Omega$).
These two views are equivalent; for $A\subseteq\Omega$ we can write either $P(A)$ or $P(x\in A)$; and for $\phi(x)$ we can write either $P(\phi(x))$ or $P(\{x\in\Omega\mid\phi(x)\})$. So it doesn't really matter which convention one uses, since it is easy to convert from one or the other.
The core definitions in textbooks will usually use the subset-of-$\Omega$ formalism, because we're familiar with using sets as mathematical objects everywhere in mathematics, whereas using a logical formula as a mathematical object that can be manipulated is somewhat of a specialty. Also, there are technical reasons (measure theory) why one might not want all subsets of $\Omega$ to be events, and such restrictions are much easier to express in a set-theoretic language than as restrictions on which logical formulas are allowed. On the other hand, it takes quite a bit of devious ingenuity to build a purely logical $\phi(x)$ that does not represent an event, so this is less of a problem in practice than it is in theory.
As you have found out, these reasons often do not stop textbook authors from using logical notation when it suits them, and expect readers to be able to translate into the set-based formalism.
As additional abuse of notation, it is not uncommon to switch between the two viewpoints in the middle of a formula, such as in $P(\neg A\land \neg B)$, where $A$ and $B$ might have been defined as sets, but are used in a context where one must have a logical formula. In this case, too, you're suspected to mentally insert the appropriate conversion, giving $P(\neg(x\in A)\land \neg(x \in B))$.