The Timpson Lyceum · Discrete Math (2025)

Probability, Random Variables, and Expectation - Lecture 16 (Discrete Math)

Overview

This lecture introduces random variables and expectation, the bridge between a raw sample space and the numerical quantities we actually care about. Despite the name, a random variable is neither random nor a variable — it is simply a function whose domain is the sample space. Once we can attach a number to every outcome, we can ask for the probability that the function takes a given value, and we can ask what value we should expect it to take on average. The lecture builds intuition through dice, coin flips, and gambling-style spinner games, connects the binomial coefficient to the probability of getting a fixed number of heads, defines independence of random variables, and closes with the definition and key properties (linearity) of expected value.

Key Concepts

Sample space. A sample space is an ordered pair $(S, P)$ where $S$ is the set of all possible outcomes and $P\colon S \to \mathbb{R}$ is the probability function assigning each outcome a number in $[0,1]$. The probabilities of all outcomes must sum to $1$. For one fair die, $S = \{1,2,3,4,5,6\}$ and $P(s) = \tfrac{1}{6}$ for every outcome.

Random variable. Given a sample space $(S,P)$ and any set $V$, a function $X\colon S \to V$ is called a random variable. The only requirement is that its domain is $S$ — it assigns something to every outcome. The defining mental model: random variable = function.

Real random variable. The case we care about almost always: a function $X\colon S \to \mathbb{R}$ that maps every outcome to a real number — just like the probability function itself. (A non-real example: a function $Z$ that maps a coin-flip string to the set of positions where heads occur. The output need not be a number; output sets can be anything.)

Probability that a random variable equals a value. Writing $P(X = a)$ means the probability of the event "the outcome maps to $a$": $P(X = a) = \sum_{\substack{s \in S \\ X(s) = a}} P(s).$ This replaces the old event-based language: instead of defining a separate event for "rolled a 7," we define one random variable $X$ and talk about $P(X = 7)$. For most real numbers $a$ (e.g. $\pi$, $27$, $-6$) this probability is $0$, since no outcome maps there.

Building new random variables from old ones. Random variables can be combined like functions. For two dice with rolls written $(a,b)$, let $X_1(a,b) = a$ and $X_2(a,b) = b$. Then $X = X_1 + X_2$, defined by $(X_1+X_2)(a,b) = X_1(a,b) + X_2(a,b) = a+b$, is the familiar "sum of the dice" random variable.

Coin flips and the binomial connection. Flip a coin $n$ times; the sample space $S^n$ consists of all length-$n$ strings of H's and T's. For a fair coin each string has probability $\tfrac{1}{2^n}$. If $X$ counts the number of heads, then the number of strings with exactly $h$ heads is $\binom{n}{h}$, so: $P(X = h) = \frac{\binom{n}{h}}{2^n} \quad (\text{fair coin}).$ For an unfair coin with $P(\text{H}) = p$ and $P(\text{T}) = 1-p$, each specific string with $h$ heads and $n-h$ tails has probability $p^{h}(1-p)^{n-h}$, so: $P(X = h) = \binom{n}{h}\, p^{h}(1-p)^{\,n-h}.$ This is exactly a term of the binomial theorem, $(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}x^{k}y^{n-k}$, with $x = p$ and $y = 1-p$. Since $p + (1-p) = 1$ and $1^n = 1$, the full sum equals $1$ — i.e. the probabilities of getting $0,1,2,\dots,n$ heads add to $100\%$, as they must.

Independent random variables. $X$ and $Y$ on the same sample space are independent provided that for all values $a, b$: $P(X = a \text{ and } Y = b) = P(X = a)\cdot P(Y = b),$ equivalently $P(X = a \mid Y = b) = P(X = a)$ — knowing $Y$ gives no information about $X$. For two dice, $X_1$ (first die) and $X_2$ (second die) are independent, but the sum $X = X_1 + X_2$ is not independent of $X_1$ (knowing the first die changes the distribution of the total).

Expected value. For a real random variable $X$ on $(S,P)$, the expected value is the probability-weighted average of its outputs: $E(X) = \sum_{s \in S} X(s)\, P(s).$ Equivalently — and usually easier to compute — sum over the values the variable can take: $E(X) = \sum_{a \in \mathbb{R}} a\, P(X = a),$ where all but finitely many terms vanish because $P(X = a) = 0$ for most $a$. (Going from the first form to the second is a regrouping of terms by output value, valid because addition is commutative and associative and because a real random variable always outputs a real number to group by.) Expectation answers "what would the long-run average be over a huge number of trials?" — e.g. one fair die has $E(X) = \tfrac{1+2+3+4+5+6}{6} = \tfrac{7}{2} = 3.5$.

Linearity of expectation. For real random variables $X, Y$ and a constant $c \in \mathbb{R}$:

• $E(X + Y) = E(X) + E(Y)$
• $E(cX) = c\,E(X)$
• combined: $E(c_1 X + c_2 Y) = c_1 E(X) + c_2 E(Y)$.

This holds regardless of independence, mirroring the linearity of limits and derivatives. It makes scaling trivial: since one die has $E = 3.5$, rolling $100$ dice has expected total $100 \times 3.5 = 350$.

Products are different. $E(XY) = E(X)\,E(Y)$ holds only when $X$ and $Y$ are independent. It does not follow from linearity (compare: the derivative of a product is not the product of derivatives — it needs the product rule).

Worked Example

Unfair-spinner gambling game. A spinner lands on $1,2,3,4$ with probabilities $\tfrac12, \tfrac14, \tfrac18, \tfrac18$. The payout random variable $X$ pays you the number you land on (in dollars): $X(k) = k$. What is the fair price to charge per spin? Compute $E(X)$.

Apply the value-weighted definition $E(X) = \sum_{a} a\,P(X=a)$, then substitute:

So the game is worth about $1.88 per spin; charging $2 turns a steady profit. Note the intuition trap: a similar "odd pays $20, even pays $10" spinner feels like a $15 game but actually has E(Y) = 20\cdot\tfrac12 + 10\cdot\tfrac14 + 20\cdot\tfrac18 + 10\cdot\tfrac18 = \text{\} 16.25$ — higher than intuition suggests.

Proof sketch — linearity $E(X+Y) = E(X) + E(Y)$. Starting from the definition and using that $(X+Y)(s) = X(s) + Y(s)$:

Summary

• A random variable is a function on the sample space $S$; a real random variable maps outcomes to $\mathbb{R}$. The name is misleading — nothing is random or variable about it.
• $P(X = a) = \sum_{s : X(s) = a} P(s)$ lets one random variable replace many separately-defined events.
• For $n$ coin flips, $P(X = h) = \binom{n}{h}p^{h}(1-p)^{n-h}$ (with $p=\tfrac12$ for a fair coin); summing over all $h$ gives $1$ by the binomial theorem.
• Random variables are independent when $P(X=a,\,Y=b) = P(X=a)P(Y=b)$; conditioning then changes nothing.
• Expected value $E(X) = \sum_{s\in S} X(s)P(s) = \sum_{a} a\,P(X=a)$ is the long-run average; for one fair die it is $3.5$.
• Expectation is linear ($E(c_1X + c_2Y) = c_1E(X) + c_2E(Y)$) for any random variables, but $E(XY) = E(X)E(Y)$ requires independence.