Life is unpredictable, and probability is our best mathematical handle on real-world uncertainty.

In this quiz, we'll use simple probabilistic models to introduce **random variables**, a key concept in quantifying chance outcomes.

A **random variable** assigns a numerical value to each outcome of a chance event. Random variables are not the same as the events they quantify.

For example, when a fair coin is flipped exactly twice, the set of all outcomes, or **sample space**, \( \Omega \) is \( \{ HH, HT, TH, TT \}, \) where H/T is short for “heads/tails”, HH is the event the coin lands heads up on both flips, HT represents a heads followed by a tails, etc.

HH, HT, TH, TT are not random variables, but the *number* of heads observed in two flips *is*. What are some other examples of random variables?

Let's take a closer look at random variables through an example.

Let \( \Omega = \{ H, T \} \) be the sample space for a single fair coin flip. H and T are not random variables, but if we assign the number 1 to H and the number 0 to T, then we've just created one: \[ X(H) = 1, \ X(T) = 0.\] \( X\) counts the number of heads in a single coin flip.

We don't know what \(X \) will be after a flip, but we can compute \( P(X=1), \) the probability that we observe \( X = 1 \) upon flipping the coin. What is \( P(X=1) ? \)

Let's make things more interesting by flipping the fair coin *three* times consecutively.

\(N,\) the number of tails in three flips, assigns a value to each element in the sample space \(\Omega:\) \[ \Omega = \{\small{ HHH, HHT, HTT, TTT, THH,TTH, HTH, THT} \}. \] For example, \( N(HTT) = 2.\) Since fair coin flips are unpredictable, \( N\) is a random variable.

Calculate \( P(N = 2).\)

**Functions** create new random variables out of more basic ones.

For example, say Xavier, Yoshi, and Zander all take a multiple-choice test by guessing; their scores are the random variables X, Y, and Z. The average \( A = \frac{1}{3} ( X + Y + Z ) \) is a function of the original three random variables and is a brand-new random variable on its own.

If the exam has three true/false questions and a score less than \( 20 \%\) doesn't pass, what is the probability the group of Xavier, Yoshi, and Zander fails on average?

So far, all of our random variables have been **discrete**, meaning their values are countable.

There are many real-world problems best modeled by a continuum of values; we associate to them **continuous random variables**.

For example, the velocity \( V \) of an air molecule inside of a basketball can take on a continuous range of values. We can't know for sure what it is, so \( V \) is a continuous random variable.

What are some other continuous random variables?

Since the number of outcomes is uncountable for a continuous random variable, we need to take a different approach to how we compute its probabilities.

We'll see in the next quiz that dealing with continuous random variables is relatively straightforward if we have a **probability distribution**.

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