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In layman terms, what is the difference between a random variable and an event? To my understanding a random variable is a function outputting a real number. And an event is an outcome or a set of outcomes.

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    $\begingroup$ Tossing a coin two times is an event with the outcome space $\{TT,TH,HT,HH\}$. A random variable maps each element of outcome space to some value. For example you could have the random variable $X$ mapping each element to the number of heads. Then $X(TT)=0,X(TH)=X(HT)=1,X(HH)=2$. $\endgroup$ Nov 2, 2020 at 6:21

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This may be an instance where the most possible general definition gives more insight. Consider a pair $(\Omega, \Sigma)$ where $\Omega$ is the sample space and $\Sigma \subseteq 2^\Omega$ is a $\sigma$-algebra. The measurable sets (i.e., elements of $\Sigma$) are called events. A random variable is a measurable function $X\colon \Omega \to \Bbb R$. By measurable function we mean that for every open interval $]a,b[\subseteq \Bbb R$, we have that $X^{-1}\big(]a,b[\big) \in \Sigma$ is measurable. To put it simply:

A random variable is a nice function from the sample space to the real line (and a random vector takes values in some $\Bbb R^k$, etc.). An event is a certain subset of the sample space, to which a probability may be assigned.

This does not depend on any choice of probability. More precisely, a probability space is a triple $(\Omega, \Sigma, \Bbb P)$ where $(\Omega, \Sigma)$ is as above and $\Bbb P\colon \Sigma \to \Bbb R_{\geq 0}$ is a measure with $\Bbb P(\Omega) = 1$. Then finding $\Bbb P(X = x)$ means finding the "measure" of the event $\{\omega \in \Omega \mid X(\omega) = x\}\in \Sigma$, inside of the sample space $\Omega$.

Example: consider the experiment of tossing a fair coin $n$ times. The sample space is $\Omega = \{{\rm heads}, {\rm tails}\}^n$. The $\sigma$-algebra of events will be all possible subsets of $\Omega$. Say that we write $0$ for heads and $1$ for tails, and the random variable $X \colon \Omega \to \Bbb R$ gives the result of the second toss. That is: $X\colon \Omega \to \Bbb R$ is given by $X(\omega_1,\ldots, \omega_n)= \omega_2$. The event "getting heads on the second toss" is $X^{-1}(0) = \{ (\omega_1,\ldots, \omega_n) \in \Omega \mid \omega_2 = 0 \}$, and the probability of this happening is $\Bbb P(X = 0) = 1/2$.

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    $\begingroup$ Good answer but I think it is a little too advanced seeing the level of the question asked and may not be comprehensible. $\endgroup$ Nov 2, 2020 at 6:31
  • $\begingroup$ I totally agree. If OP gets the take-away "A random variable is a nice function from the sample space to the real line (and a random vector takes values in some $\Bbb R^k$, etc.). An event is a certain subset of the sample space, to which a probability may be assigned." then it's already good enough for me. $\endgroup$
    – Ivo Terek
    Nov 2, 2020 at 6:47
  • $\begingroup$ @ShubhamJohri subtle flex :^) $\endgroup$ Sep 26, 2021 at 6:28
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We have a large (and maybe not well describable) space $\Omega$ of possible outcomes. For example, $\Omega$ might be the set of possible weathers tomorrow at the village where I live.

An event is a decent subset of $\Omega$, say the set $R\subset\Omega$ of all weathers where it rains at time 10:00 tomorrow. When a probability measure ${\tt P}$ has been defined on $\Omega$ it makes sense to talk of the probability ${\tt P}(R)$ of the event $R$, and the well known rules about the probabilities of unions, etc., of events hold.

A random variable is a given real-valued function defined on $\Omega$, say, the temperature at the graveyard at time 12:00. This is a nice function, and there is nothing random about it. When fate has chosen a weather $\omega\in\Omega$ the function value $f(\omega)$ is determined. This value is considered "random", before one has really observed it.

Usually the probability that $f$ has a particular value, say $21.35^\circ$ Celsius, is zero. This particular event is just too special. But one can easily define events using $f$ that have an interesting probability, say the event $Z$, consisting all weathers $\omega$ with $f(\omega)<0^\circ$ Celsius. We see here that a random variable can be used to create lots of events that are of interest in the momentary discussion.

On the other hand an arbitrary event, say the $R$ from above, can be redesigned as a random variable, when we want to argue in the language of random variables: The characteristic function of $R$, being $\equiv1$ on $R$ and $\equiv0$ outside of $R$ is a perfect random variable, albeit one with just two values.

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