Expected value (EV) answers one question: if you could repeat this bet many times under the same probabilities, what do you earn per try on average? Module 01 maps price to belief; Module 02 charges you the ask; arbitrage pays when sums misprice aligned payoffs. EV is the north star for whether a single trade is worth making—before Kelly sizing and variance stress in the chapters that follow.
Definition
For discrete outcomes with probabilities p_i and net payoffs x_i, EV is the weighted sum p_i × x_i. On a $1 binary YES bought at price c, if YES happens you net (1 − c); if NO happens you net (−c). With your model probability p for YES, EV per share equals p(1 − c) + (1 − p)(−c), which simplifies to p − c. Edge in probability points is p minus your execution price when you are the buyer.
That identity is why traders say “I have six points of edge” when they believe 55% and pay 49¢.
Kalshi YES at the ask
“Rate cut by June” with YES ask 44¢ and your model p = 52%: EV per share = 0.52 − 0.44 = +$0.08. On 1,000 shares, expected profit is $80 before fees. A taker fee of roughly a cent per contract might shave $10–$20 depending on schedule—EV after fees is what matters.
Buying NO instead of economically shorting YES: if YES ask is 44¢, NO ask might be 56¢. For NO, p(NO) = 0.48 and EV(NO) = 0.48 − 0.56 = −$0.08—same negative edge. Use the leg with the better ask and the probability on that leg.
Spread and Polymarket
Display 47% mid with YES ask 49¢ and bid 45¢; your p = 55%. Buy YES at 0.49: EV = 0.55 − 0.49 = +$0.06 per share. Selling YES at the bid uses a different payoff tree—always write the leg you actually trade. Long YES EV = p − ask; long NO EV = (1 − p) − ask_NO.
Arbitrage EV
Risk-free arb might buy YES on venue A at 48¢ and sell YES on venue B at 55¢ for +7¢ per matched pair if resolution is identical—probability one modulo counterparty and rule risk. Structural basket arb sums legs; EV is additive when joint probabilities are coherent. Different resolution strings destroy the “probability one” story.
Partial fills leave unhedged binary risk; withdrawal delay is capital path risk, not a formula issue.
Fees and when to pass +EV
Taker percent on wins, per-contract fees, gas on-chain, and spread (already in c if you use ask/bid) all shrink numerator edge. A toy with 2% taker on the win branch can kill trades that looked +3¢ naked.
Positive EV is not a must-bet command. Variance can make +5¢ EV on a huge fraction of bankroll reckless. Model error swallows edge inside confidence intervals. Liquidity may not fill size at the ask. Correlation stacks exposure on the same factor. Resolution risk means Ω mismatch, not mispricing.
Expected value of information: before paying for a poll, ask whether the market already moved on the same feed. If Kalshi reprices instantly on public data, residual EV of buying that data ≈ 0.
Multi-outcome and breakeven
Buy candidate B at 28¢ with p_B = 35%: EV = +0.07. Still verify your full vector sums to one and that buying every outcome at asks is not negative EV by construction.
Breakeven for YES bought at 62¢ needs p = 62% before fees; add ~2¢ effective cost and you need about 64%. Entry at 40¢ needs 40%; at 75¢ needs 75%.
Comparison of edge types
A better forecast expresses edge as p > ask. Locked arb expresses edge as sum of payoffs minus costs. Maker rebates add to p − c when you provide liquidity. Fading manipulation uses your p versus a spiked mid—only after you confirm thin liquidity, not information.
Hygiene before you click
State payoff ($1 binary default). Write your p with timestamp and reason. Record c as ask (buy) or bid (sell). EV per share = p − c for long YES. Multiply by intended size; subtract fees. Note portfolio correlation with open positions. Compare to a minimum edge rule (e.g. three cents plus fees). Log outcome for Brier later.
Kalshi: YES ask, bid on exit, published fee schedule. Polymarket: pool price plus slippage table, gas, protocol fee. EV must use executable c, not a screenshot mid.
Common mistakes
EV from mid overstates edge by half the spread. Confusing ROI with EV: 8¢ on 44¢ is 18% ROI but still 8¢ per share. Double-counting arb across different resolutions. Ignoring that NO may be the better leg when YES spread is wide.
Maker rebate and negative fees
Some venues rebate makers. A +2¢ rebate on a fill where your model edge is +1¢ can flip net EV positive—but only if you actually provide liquidity and get filled at your quote. Taker strategies should not count maker rebates you never earn.
Scaling EV to the portfolio
Per-share EV times shares minus fees is dollar EV for one line. Open ten correlated lines and dollar EV adds, but risk does not diversify the way ten coin flips would. The correlation chapter explains why portfolio EV can look healthy while one election night erases the year—EV is linear in expectation, not in comfort.
Minimum edge rule of thumb
Many discretionary traders demand edge > 2× (spread half-width + fees) before clicking. On a 4¢ wide market with 1¢ fee drag, that might mean p − ask > 5¢. The threshold is personal; the logic is universal: small edges die first when variance and model error are real.
Selling YES EV in words
You receive bid b now and owe $1 if YES wins. With belief p, expected value of a short YES position is (1 − p) − (1 − b) = b − p—mirror of long YES. If p = 55% and bid 58¢, selling YES has EV +3¢ per share. Traders forget the short formula and only compute long YES.
Information already in price
If your posterior after a public poll equals 62% and the ask was 61% before the poll and 63% after, your residual edge may be zero—the market already paid for the information. EV of trading the headline is not EV of having the headline unless you were faster than the book.
Categorical EV sketch
Buy candidate C at 22¢ with belief 30%: EV = +8¢ per share if rules are a single winner. Also compute whether your implied joint over other candidates is coherent—EV on one leg without a full vector is half a model. Module 05 develops categorical payoffs; EV logic starts here.
Zero EV as information
When p ≈ c after fees, EV ≈ 0—the market and you agree. That is not a failure; it is the sign to pass or to improve data. Discretionary edge lives in the gaps, not in forcing trades to feel productive.
Worked arb EV in one paragraph
Buy YES here at 48¢, sell YES there at 55¢ on matched rules: locked 7¢ per pair minus transfer and fee. EV is not p − c on either leg—it is the wedge. Do not apply binary long formulas to locked packages; add leg cash flows instead. If one leg fails to fill, revert to single-contract EV with full variance.
Subjective p is a distribution
Your p is a point estimate; honest forecasters carry uncertainty. If p might be 52–58% and the ask is 54¢, EV is fragile. Widen required edge when the model is noisy. The Kelly chapter shrinks size for the same reason.
What comes next
Next: variance and standard deviation—why a +EV bet can still be wrong for your bankroll on one resolution.