Comparing Options with Expected Value — I can compare two options (games, decisions) using expected value and justify a choice at my level. - Math 3

SECTION 1

Historical Context & Motivation

People have been making risky decisions for millennia — from ancient dice games to modern stock markets. But for most of history, there was no mathematical framework for comparing uncertain choices. You relied on gut instinct, superstition, or trial and error. The idea that you could calculate which option is better, even when both involve chance, was a genuine breakthrough. That breakthrough is the concept of expected value, and it transformed gambling, insurance, economics, and everyday decision-making.

1654 • The Problem of Points

Blaise Pascal and Pierre de Fermat exchanged letters about how to fairly divide stakes in an interrupted dice game. Their correspondence laid the groundwork for probability theory and the notion of expected winnings.

1657 • Huygens Formalizes "Expectation"

Dutch mathematician Christiaan Huygens published the first formal treatment of expected value in De Ratiociniis in Ludo Aleae, giving mathematicians a precise formula for comparing uncertain outcomes.

1713 • Bernoulli's Ars Conjectandi

Jacob Bernoulli's posthumous work connected expected value to the Law of Large Numbers, proving that observed averages converge to the theoretical expected value over many repetitions.

1944 • Game Theory Arrives

John von Neumann and Oskar Morgenstern published Theory of Games and Economic Behavior, using expected value as the backbone of rational decision-making under uncertainty.

The central question that expected value answers is deceptively simple: When two options both involve uncertainty, how do I decide which one is better in the long run? Whether you are choosing between two carnival games, two business strategies, or two insurance plans, expected value gives you a single number for each option so you can compare them directly.

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Core Principles & Definitions

Before you can compare options, you need to understand what expected value actually measures. Think of it as the long-run average outcome you would experience if you repeated the same decision over and over. It combines every possible outcome with how likely that outcome is, condensing all of that information into one meaningful number.

Outcome

A specific result that can happen. For a dice roll, each face (1 through 6) is a separate outcome. For a game, an outcome is usually a dollar amount you win or lose.

Probability

A number between 0 and 1 (or 0% and 100%) that measures how likely a particular outcome is. The probabilities of all possible outcomes must add up to exactly 1.

Expected Value (EV)

The weighted average of all possible outcomes, where each outcome is weighted by its probability. A higher EV means a better long-run average, making it the key tool for comparison.

Fair Game

A game or decision with an expected value of exactly $0. Over time, you would neither gain nor lose money on average. Most real games are not fair — one side has an edge.

Comparing Options

To compare two options, compute the EV of each one separately. The option with the higher (or less negative) expected value is the mathematically better choice in the long run.

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KEY TAKEAWAY

Think of expected value like a GPS that shows average travel time. One route might sometimes be faster and sometimes slower, but the GPS shows the average. Similarly, EV does not tell you what will happen on any single play — it tells you what the average result would be over hundreds of plays. When comparing two routes (or two games), pick the one with the better average.

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Visualizing Expected Value Comparisons

The diagram below shows two carnival games side by side. For each game, the possible outcomes are shown as bars whose height represents the dollar value and whose width represents the probability. The expected value for each game is marked with a dashed line, letting you instantly see which game has the higher long-run payoff.

Game A offers a 50% chance to win $4 and a 50% chance to lose $1, yielding an EV of $1.50. Game B offers a 10% chance to win $8 and a 90% chance to lose $1, yielding an EV of −$0.10. Even though Game B has a bigger possible win, Game A is the better long-run choice because its expected value is higher.

Notice how the bar heights represent dollar values and the labels underneath show probabilities. Game B has a tall, impressive bar for the $8 win, but that outcome only happens 10% of the time. The other 90% of the time you lose $1. When you average everything out, Game B actually costs you money in the long run. Game A's outcomes are more balanced, and the math works in your favor. This is why expected value beats intuition — the flashier prize is not always the smarter bet.

SECTION 4

Mathematical Framework

Computing expected value is straightforward once you know the formula. You multiply each possible outcome by its probability, then add all those products together. This gives you the weighted average that represents your long-run result.

EXPECTED VALUE FORMULA

E(X) = x₁ · P(x₁) + x₂ · P(x₂) + ⋯ + xₙ · P(xₙ)

Where x₁, x₂, …, xₙ are the possible outcomes (dollar amounts, points, etc.) and P(x₁), P(x₂), …, P(xₙ) are their respective probabilities. Remember: all probabilities must sum to 1.

COMPACT NOTATION (SUMMATION)

E(X) = Σ [ xᵢ · P(xᵢ) ] for i = 1 to n

This is the same formula written with summation notation. Each term xᵢ · P(xᵢ) represents one outcome multiplied by its probability, and Σ means "add them all up."

COMPARISON RULE

Choose Option A over Option B when E(A) > E(B)

After computing the expected value for each option, the option with the higher expected value is the mathematically better choice. If both are negative, the less negative value loses you less money on average.

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Watch the Signs

Losses should be entered as negative numbers. If you pay $2 to play a game and can win $5 back (net gain of $3) with probability 0.30 and lose your $2 (net loss of −$2) with probability 0.70, use x₁ = +3 and x₂ = −2 in the formula. Forgetting the negative sign is the most common mistake.

SECTION 5

Mapping Decisions with a Tree Diagram

When a decision has multiple branches and outcomes, a decision tree is the clearest way to organize the information before you calculate expected value. Each branch shows an outcome, its probability, and its value. By working from the tips of the branches back toward the trunk, you accumulate the expected value for each option and can compare them visually.

The decision tree maps both games from a single decision point. Each branch shows the probability and the resulting product. Adding the products within each game gives the expected value on the right side.

Reading the tree from left to right, you start at the decision node (the rectangle), branch into each game (the circles are chance nodes), and end at the outcome rectangles on the right. The small calculations beside each outcome show each product, and the shaded boxes total them. This approach is especially powerful when decisions have three or more outcomes, because the tree keeps everything organized and makes it nearly impossible to forget an outcome.

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Worked Example: Choosing Between Two Raffles

Your school sells tickets for two different raffles. Each ticket costs $5. Raffle A has 100 tickets and offers a single prize of $200. Raffle B has 50 tickets and offers a single prize of $80. You can only buy one ticket total. Which raffle gives you a better expected value?

Raffle Comparison

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Step 1 — Identify outcomes and net values

In both raffles, you pay $5 for a ticket. If you win Raffle A, you receive $200, so your net gain is $200 − $5 = $195. If you lose, your net result is −$5. For Raffle B, winning gives $80 − $5 = $75 net, and losing gives −$5.

Raffle A outcomes: +$195 or −$5. Raffle B outcomes: +$75 or −$5.
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Step 2 — Assign probabilities

Raffle A has 100 tickets, so your probability of winning is 1/100 = 0.01, and losing is 99/100 = 0.99. Raffle B has 50 tickets, so winning is 1/50 = 0.02, and losing is 49/50 = 0.98.

P(win A) = 0.01, P(lose A) = 0.99; P(win B) = 0.02, P(lose B) = 0.98
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Step 3 — Calculate E(A)

Apply the expected value formula to Raffle A: E(A) = (195)(0.01) + (−5)(0.99) = 1.95 + (−4.95) = −3.00.

E(A) = −$3.00
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Step 4 — Calculate E(B)

Apply the formula to Raffle B: E(B) = (75)(0.02) + (−5)(0.98) = 1.50 + (−4.90) = −3.40.

E(B) = −$3.40
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Step 5 — Compare and justify

Both expected values are negative, which makes sense — the organizers need to make money. However, −$3.00 > −$3.40, so Raffle A loses you less money on average per ticket. If you must buy a ticket, Raffle A is the better choice because its expected value is closer to zero.

Raffle A is the better option since E(A) = −$3.00 > E(B) = −$3.40.

SECTION 7

Strengths and Limitations of Expected Value

Expected value is a powerful tool, but like any model, it has boundaries. Understanding what it does well — and where it falls short — makes you a more thoughtful decision-maker.

Strengths	Limitations
Reduces complex, multi-outcome scenarios to a single comparable number.	Tells you the long-run average but says nothing about risk or variability on a single play.
Works for any number of outcomes as long as you know the probabilities and values.	Requires accurate probabilities — if your estimates are wrong, your EV is wrong too.
Objective and mathematical, removing emotional bias from decision-making.	Ignores personal factors like how devastating a loss would be (e.g., risking rent money).
Applies across domains: games, business, medicine, daily life choices.	Most useful for repeatable decisions; less meaningful for one-time, irreversible choices.

Strengths and limitations of using expected value for comparing options.

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WHEN EV ISN'T ENOUGH

Imagine two job offers: Job A pays $50,000 guaranteed, and Job B pays $0 half the time and $110,000 the other half. Job B has a higher expected value ($55,000), but if you can't afford to earn $0, Job A might be the wiser choice. This is why economists developed expected utility — a related concept that accounts for how much a person values (or fears) extreme outcomes. For now, just know that EV is the foundation, and real-world decisions sometimes need an extra layer of judgment on top of it.

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Connection to Advanced Probability & Decision Theory

Expected value is the starting point of a much larger landscape. As you advance in mathematics and statistics, you will encounter tools that build directly on EV, adding nuance for more complex situations.

Concept	What It Adds Beyond EV	Where You'll See It
Variance / Standard Deviation	Measures how spread out outcomes are around the EV. Two options can have the same EV but very different levels of risk.	AP Statistics, college probability, portfolio theory in finance.
Expected Utility Theory	Replaces dollar values with "utility" values that reflect personal risk tolerance. Risk-averse people assign lower utility to gambles even when EV is positive.	Microeconomics, behavioral economics, decision science.
Conditional Expected Value	Computes EV based on new information. If you learn something midway through a game, you can update your expected value accordingly.	Bayesian statistics, machine learning, medical diagnosis.
Game Theory (Nash Equilibrium)	Extends EV to situations where another person is also making strategic choices. Both players compute EV while anticipating each other's moves.	Economics, political science, evolutionary biology.

How expected value connects to more advanced topics.

The key point is that mastering EV now gives you the algebraic and conceptual foundation for all of these advanced topics. Every one of them starts with the same fundamental idea: multiply outcomes by probabilities and add them up. The extensions just add more sophisticated ways to handle risk, information, and human psychology.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Game X has an expected value of $2.50, and Game Y has an expected value of −$0.75. A friend says, "Game Y must be better because last time I played it, I won $20!" Explain, using the idea of expected value, why your friend's reasoning is flawed and which game you should prefer in the long run.

Show answer

Your friend's experience was a single trial, but expected value describes the average outcome over many trials. One lucky win does not change the mathematical average. Because E(X) = $2.50 is greater than E(Y) = −$0.75, Game X will, on average, earn you $2.50 per play while Game Y will cost you $0.75 per play. Over many repetitions, Game X is the superior choice regardless of any single outcome your friend observed.

Question 2: BASIC CALCULATION

A spinner has three sections: red (P = 0.50, win $3), blue (P = 0.30, win $1), and green (P = 0.20, lose $4). Calculate the expected value of one spin.

Show answer

Apply the expected value formula: E(X) = (3)(0.50) + (1)(0.30) + (−4)(0.20) = 1.50 + 0.30 + (−0.80) = $1.00. The expected value is +$1.00, meaning you gain an average of one dollar per spin in the long run. Since the EV is positive, this game is favorable for the player.

Question 3: INTERMEDIATE

You must choose between two games. Game M: roll a standard die; win $12 if you roll a 6, lose $2 otherwise. Game N: flip two coins; win $5 if both are heads, win $1 if exactly one is heads, lose $3 if both are tails. Compute the expected value of each game and determine which is the better choice.

Show answer

For Game M: P(6) = 1/6, P(not 6) = 5/6. E(M) = (12)(1/6) + (−2)(5/6) = 2.00 + (−1.667) = $0.33 (approximately). For Game N: P(HH) = 1/4, P(exactly one H) = 2/4 = 1/2, P(TT) = 1/4. E(N) = (5)(0.25) + (1)(0.50) + (−3)(0.25) = 1.25 + 0.50 + (−0.75) = $1.00. Since E(N) = $1.00 > E(M) ≈ $0.33, Game N is the better option. Both games have positive EVs, but Game N returns roughly three times more per play on average.

Question 4: APPLIED

A delivery company can choose between two shipping routes. Route A has a 90% chance of on-time delivery (earning a $50 bonus) and a 10% chance of being late (incurring a $200 penalty). Route B has a 70% chance of arriving early (earning a $70 bonus), a 20% chance of on-time (earning a $50 bonus), and a 10% chance of being late (incurring a $200 penalty). Which route should the company choose based on expected value?

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For Route A: E(A) = (50)(0.90) + (−200)(0.10) = 45 + (−20) = $25. For Route B: E(B) = (70)(0.70) + (50)(0.20) + (−200)(0.10) = 49 + 10 + (−20) = $39. Route B has a higher expected value ($39 vs. $25), so the company should prefer Route B. The 70% early-arrival probability combined with the larger $70 bonus provides enough extra income to outweigh the risk, which is identical between the two routes.

Question 5: CRITICAL THINKING

Suppose Game P has E(P) = $5 with outcomes ranging from −$100 to +$110, and Game Q has E(Q) = $4 with outcomes ranging from +$3 to +$6. A student says, "Always pick the option with the higher expected value, so Game P is always better." Critique this statement. Under what circumstances might a rational person prefer Game Q despite its lower expected value?

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The student's rule works perfectly for long-run, repeatable decisions with no concern about individual outcomes. However, Game P carries the risk of losing $100 on a single play, which could be devastating if you cannot afford that loss. Game Q, by contrast, guarantees at least $3 — it has virtually no risk. A person who can only play once, or who cannot absorb a $100 loss, might rationally prefer Game Q because the certainty of a positive result outweighs the slightly higher average of Game P. This reasoning connects to expected utility theory: the pain of losing $100 may far outweigh the pleasure of gaining an extra $1 on average. EV is a necessary starting point, but it does not capture risk tolerance or real-world constraints.

SUMMARY

Lesson Summary

Summary

Expected value is the weighted average of all possible outcomes, found by multiplying each outcome by its probability and summing the products: E(X) = Σ[xᵢ · P(xᵢ)]. To compare two options, compute the EV of each one and choose the option with the higher value. Remember to use negative numbers for losses and to verify that all probabilities sum to 1 before computing.

While expected value is the cornerstone of rational decision-making under uncertainty, it does have limitations. It describes the long-run average rather than what will happen on any single trial, and it does not account for risk tolerance. Tools like variance and expected utility build on EV to handle those additional complexities. Mastering EV now gives you a solid foundation for all of these advanced ideas.