Do the states of a Random Variable have to be Mutually Exclusive? Say I'm interested in the probability of a person's career. In a hypothetical world, people can only be: {doctors, lawyers, teachers}. If I wanted to make career a random variable, I'd need to map these categories onto real numbers. But, do the 3 categories have to be mutually exclusive in order to be a "proper" random variable?
That is, in the hypothetical world, let's say someone could be a teacher and a doctor. Is it no longer ideal to think of career as a random variable?
Do the values of a random variable have to be mutually exclusive or disjoint?
 A: A random variable $X$ is a function from a sample space $\Omega$ to a state space $S$. In this case $S = \{\text{doctor, lawyer, teacher}\}$. (Technically, $S$ must be a measurable space, but this requirement is easily accommodated.)

Do the values of a random variable have to be mutually exclusive or disjoint?

The answer is no, but this may not be your question. Since $X$ is a function, $X(\omega)$ is a single value for each $\omega \in \Omega$. We cannot have $X(\omega) = \text{doctor}$ and $X(\omega) = \text{lawyer}$ for the same $\omega$. On the other hand, you could have $X(\omega) = \text{lawyer}$ for multiple $\omega \in \Omega$, so the sets $\{X(\omega_1)\}$ and $\{X(\omega_2)\}$ need not be disjoint even for $\omega_1 \neq \omega_2$.
Finally, you can accommodate being both a teacher and a doctor by defining a fourth element in $S$ which is $\text{doctor and teacher}$. Then $S = \{\text{doctor, lawyer, teacher, doctor and teacher}\}$, and  the career function $X$ can have positive probability for the state $\text{doctor and teacher}$.
A: By definition, a random variable is a function of events in a sample space. In this case, the sample space is $\{\text{doctor,lawyer,teacher}\}$. You could then define a random variable $\text{career}$ to have a value between $1$ and $8$, with each number corresponding to a unique combination of careers. For example, doctor and lawyer could correspond to $6$.
