Balance scale of outcomes

Expected value is the probability-weighted average of all possible numerical outcomes. For an immediate, undiscounted game, a risk-neutral fair entry price equals expected gross payout. This is not a general financial pricing rule: a replicable future payoff is priced by its replication cost, equivalently its discounted risk-neutral expectation under the model assumptions, rather than its physical-probability average. Because expectation is linear, the expected value of combined bets is simply the sum of their individual EVs, even if the bets are correlated. Under formula scoring with negative marking, evaluating expected value determines when guessing is mathematically optimal.

Words you need

Expected value (EV): The average result if you could repeat the bet many times: each outcome times its probability, added up.e.g. A die pays its face in dollars: EV = 3.5.
Edge: Your expected profit per trade or bet. Positive edge means you win on average.
Mark: Valuing a position at a reference price, usually the mid, before you actually sell it.

Why it works

For independent, identically distributed trials with finite expected absolute payoff, the sample average converges to the probability-weighted expected payoff.

EV = sum(p_i * x_i)

Spot it: You need to price a game, roll, or bet before committing capital.

Technique reference and sources

Picture a balance scale where every possible payoff sits along a beam at its dollar value. Each outcome places a weight equal to its probability on that point. The expected value is the exact balance point (center of mass) of the beam. A fair die paying $1 to $6 balances at 3.5. A bet winning $10 with probability 0.3 and losing $4 with probability 0.7 balances at 0.3 x 10 - 0.7 x 4 = 3 - 2.8 = 0.2.

When this method fits

You need to price a game, roll, or bet before committing capital.
You must decide whether to guess or omit under negative marking penalties.
You want to evaluate combined bets quickly using linearity of expectation.
A wager offers a large top prize but a low probability that may drag overall EV negative.

List gross payoffs. Identify every distinct outcome and its gross payoff before any entry cost. Fair entry price equals expected gross payout for this immediate, risk-neutral game.
Assign probabilities. Determine the probability of each outcome, checking that probabilities sum to 1.
Weight each outcome. Multiply each gross payoff by its probability: p_i x x_i.
Sum and compare. Add all weighted payoffs for the expected gross payout. Expected profit equals expected gross payout minus entry price; if profit > 0, the game has positive edge.

General guessing threshold: With reward R > 0, loss L > 0 and skip score 0, answering has EV = pR - (1-p)L. To maximize expected score, answer if p > L/(R+L); at equality answering and skipping tie. This excludes time cost and score-dependent objectives.

+1/-0.25: p > 0.2
+1/-1 (the Flow-style mock): p > 0.5; four uniform remaining choices give -0.5 EV

Uniform discrete average: When consecutive integer payouts 1 through n are equally likely, EV is the arithmetic mean of endpoints.

6-sided die paying face: (1 + 6) / 2 = 3.5
4-sided die paying face: (1 + 4) / 2 = 2.5

Binary win-loss bet: Multiply win amount by win probability, multiply loss amount by loss probability, and subtract.

0.3 x 10 - 0.7 x 4 = 3 - 2.8 = 0.2
0.2 x 20 - 0.8 x 3 = 4 - 2.4 = 1.6

Formula scoring with elimination: For a 5-choice question with +1 correct, -0.25 wrong and 0 skipped, uniform blind guessing has EV = 0; eliminating 1 option leaves 4 choices with EV = 0.25 - 0.75 x 0.25 = 0.0625.

Blind 5 choices: 0.2 x 1 - 0.8 x 0.25 = 0
1 eliminated: 0.25 x 1 - 0.75 x 0.25 = 0.0625

Common traps

Leaving a multiple-choice question blank after eliminating one option on a test with negative marking (+1 / -0.25). Only for this +1/-0.25, skip-0, five-choice example, eliminating one certainly wrong option gives uniform-guess EV +0.0625. Other penalties require p > L/(R+L); with +1/-1, p must exceed 0.5.

Multiplying the gain by its probability but subtracting the full loss without weighting it. Weigh both sides by their probabilities: EV = p x Win - (1 - p) x Loss. In a 0.3 win of $10 and $4 loss, subtract 0.7 x 4 = 2.8, not 4.

Assuming linearity of expectation requires outcomes to be independent. For integrable random variables (finite expected absolute values), E[X + Y] = E[X] + E[Y], whether independent or correlated. Undefined expectations cannot be manipulated this way.

Another example: Guessing under negative marking after eliminating 1 of 5 choices

Remaining choices: 5 - 1 = 4
Probability of guessing correctly: 1/4 = 0.25
Probability of guessing incorrectly: 3/4 = 0.75
Expected value: 0.25 x 1 - 0.75 x 0.25 = 0.25 - 0.1875 = 0.0625
Because 0.0625 > 0, guessing is mathematically superior to leaving the question blank

Source notes

These explanations and questions are written for QuantArc. Links show the underlying mathematics and learning design.

Harvard Stat 110: Probability (Joe Blitzstein)
Free course site with videos and the open companion book Introduction to Probability; the topic spine for counting, Bayes, and linearity. Lesson examples and questions are original.

Budescu & Bar-Hillel, To Guess or Not to Guess: A Decision-Theoretic View of Formula Scoring (1993)
Decision-theoretic analysis of formula scoring and rational guessing after eliminating distractors. Lesson examples and numbers are original.

Xinfeng Zhou, A Practical Guide to Quantitative Finance Interviews (2008)
Bibliographic reference for interview preparation; the library catalogue identifies the book, but does not verify its chapter methods or these lessons' fair-value and odds claims. Full book text has not been reviewed. Lesson examples and questions are original.

IES: Organizing Instruction and Study to Improve Student Learning
Supports worked examples, retrieval, explanations and spaced study. Does not establish QuantArc's exact lesson length, mastery threshold or review intervals.