Let's take a closer look at some terminology. The plain language we use above can be translated into mathematical language through the use of sets and functions. - The set of all possible outcomes is the *sample space*. - An *event* is a subset of all possible outcomes. ### Mathematical definitions >[!NOTE] Random Variable (Mathematical Definition) >A random variable $X$ is a function whose domain is the sample space $\Omega$ and whose range is subset set of real numbers $\mathbb{R}$. >If the range of $X$ is a finite (or countable) set of values, we call $X$ *discrete*. >If the range of $X$ is an interval (or countable union of intervals), we call $X$ *continuous*. ##### Example: Flipping three two-sided coins Suppose we have three coins, each with a side we call "heads" ($H$) and a side we call "tails" ($T$). **Experiment** Flip each coin and record whether each is an $H$ or a $T$. **Outcomes** The sample space for the experiment can be written as the following set $ \Omega = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}. $ **Examples of events** We could say, "let $A$ be the event that there is exactly one $H$," and "let $B$ be event that there are the second flip was an $H$. Then $ A = \{HTT, THT, TTH\} \text{ and } B = \{HHH, HHT, THH, THT\}. $ **Examples of random variables** It is somewhat natural to say, "let $X$ be the number of heads in the sequence." Why do mathematicians say that $X$ is a function? Well, consider that for each element of the sample space $\Omega$, there is an assigned numerical value. $ \begin{aligned} X(HHH) &= 3 \\ X(HHT) &= 2 \\ X(HTH) &= 2 \\ X(THH) &= 2 \\ X(HTT) &= 1 \\ X(THT) &= 1 \\ X(TTH) &= 1 \\ X(TTT) &= 0 \end{aligned} $ As we explore different