Sportsbooks quote decimal 2.50, fractional 3/2, or moneyline +150. Kalshi prints 47¢. Polymarket shows 47%. Module 01 treats these as probability dialects; Module 02 reminds you that you pay the ask; Module 03 hunts when two dialects disagree after fees and rules align. This chapter is translation—plus vig, the hidden tax that makes naive conversion lie.
Three odds formats
Decimal (European) 2.50 means total return 2.50 per unit staked including stake—profit 1.50 if you win. Fractional 3/2 means profit 1.50 per 1.00 staked. American moneyline +150 means profit 150 per 100 risked; −200 means risk 200 to win 100.
Fair implied win probability without vig: for decimal d, P = 1/d. For fractional a/b, P = b/(a+b). For positive ML +x, P = 100/(x+100). For negative ML −x, P = |x|/(|x|+100).
Same edge, different dresses
If your true chance is 40%, fair decimal is 1/0.40 = 2.50, fractional about 3/2, moneyline about +150. If the book offers decimal 2.20, raw implied P = 1/2.20 ≈ 45.5%—the book thinks the team is stronger than you do. Whether to back it requires vig math below.
At 25% true chance, fair decimal is 4.00 (ML about +300). At 60%, fair decimal is about 1.667 (ML near −150). Prediction market natives should think in P, quote in cents, and verify in decimal when talking to sports bettors.
Moneyline pair and overround
Two-way moneyline with no draw: favorite −180 implies 180/280 ≈ 64.3%; underdog +155 implies 100/255 ≈ 39.2%; sum 103.5%. The excess over 100% is overround or vig.
Proportional devig divides each raw percentage by the sum: 64.3/103.5 ≈ 62.1% and 39.2/103.5 ≈ 37.9%. Sophisticated books use Shin or power methods; proportional devig is enough to screen arbitrage, not to claim the last basis point.
Kalshi and Polymarket
Kalshi YES ask 47¢ on a $1 contract is like paying decimal 1/0.47 ≈ 2.13 for YES if fee-free: you risk 0.47 to win 0.53 net. When you sell YES, use the bid. If NO ask is 55¢, 0.47 + 0.55 = 1.02 overround—same idea as the sportsbook sum. Polymarket 0.47 USDC on a $1 share maps the same before gas and pool fees.
Devig workflow in prose
Convert each outcome to raw implied percent. Sum them. Overround equals sum minus 100%. If sum exceeds 100%, divide each leg by the sum for a quick no-vig line. Compare that line to Kalshi or Polymarket executable prices. Subtract fees mentally as a worse effective decimal before you size trades.
Three-way soccer 1X2 markets sum three decimals—105–110% is common. Draw contracts break simple moneyline-to-binary mapping; write down Ω first.
Cross-venue screen
Candidate wins nomination: Kalshi YES ask 52¢ → 0.52. Rival quoted offshore at decimal 2.40 on the opponent → rival implied 1/2.40 ≈ 0.417, so your side might be ≈ 0.583 if the book is exhaustive. Polymarket YES 49¢. A gap tempts buy Poly / sell Kalshi—but first match resolution (same primary, delegates, date), timing of settlement, fees and gas, and capital rails. Three to eight cents often is vig plus rule mismatch, not gift money.
American ↔ decimal anchors
+100 is decimal 2.00 and raw 50%. +200 is 3.00 and 33.3%. −110 is about 1.909 and 52.4% raw (still vigged). −200 is 1.50 and 66.7% raw. Sign errors on American favorites flip implied P wildly.
Odds versus probability framing
Odds-native traders speak edge as “getting 2.80 when fair is 2.50.” Probability-native traders say “my 40% versus market 35.7%.” Stake thinking: win 1.5 units versus risk 0.47 to win 0.53. Arb thinking: back 2.10, lay 2.05 elsewhere, or sum devigged probabilities below 100% when rules match.
Pitfalls
Ignoring vig—“decimal 2.0 on both sides” cannot be fair. Using favorite −110 as your true 52.4% without devig. Comparing moneyline to prediction markets without normalizing overround. Using mid on AMM pools after conversion—spread discipline from Module 02 still applies.
Effective decimal after fees
If you must pay 2¢ round-trip on a 50¢ Kalshi contract, your break-even belief is not 50% but slightly higher on YES buys and slightly lower on YES sells. Think of fees as shifting the decimal you need: a +150 fair line becomes worse once juice is explicit. Prediction markets publish fees in cents; sportsbooks hide them in overround—both belong in the same mental spreadsheet.
Narrative comparison: poll vs market vs book
| Source | What it measures | Update speed | Weak spot |
|---|---|---|---|
| Opinion poll | Stated intent at a snapshot | Weeks | Sample bias, non-response |
| Sportsbook line | Price with vig | Minutes | Overround, different event |
| Prediction market | Risk capital at touch | Continuous | Thin liquidity, rule drift |
Convert all three to devigged P before you declare disagreement. A poll at 52% and a market at 48¢ are not automatically four points of edge—the poll is not a contract and the market charges spread.
When conversion is enough
Conversion algebra does not replace Module 03 hygiene. Use odds translation to screen; use resolution matching, fee budgets, and fill simulation to trade. If devigged book probability is 58% and Kalshi ask is 52¢, you still need a model for whether the book and Kalshi describe the same Ω and whether you can size at the ask.
Fractional odds in prose
Fractional 5/1 means profit five per one staked—fair probability 1/6 ≈ 16.7%. Fractional 1/5 means profit one per five staked—fair probability 5/6 ≈ 83.3%. UK traders speak fractional; U.S. contracts speak cents. Convert both to P before you argue with a friend who watches soccer books while you watch Kalshi.
Implied probability from payoff multiples
If you risk c to win (1 − c) on a $1 binary, the market’s break-even belief for buyers is c (ignoring fees). Decimal for the YES buyer is 1/c. At 25¢, decimal 4.0 and implied 25%. At 75¢, decimal 1.33 and implied 75%. That inversion is the bridge between Module 01’s cents and a sportsbook’s +300.
Lay versus back in prediction markets
Sports bettors back an outcome (buy) or lay it (sell to others). Selling YES on Kalshi at the bid is laying YES in spirit—you offer the other side a price. Convert both sides to P before comparing to a book that quotes only the favorite. Symmetry prevents you from hunting edge on only one verb.
Rounding and display
Polymarket 47% and Kalshi 47¢ are the same idea on a $1 face; rounding to whole percents hides a 0.5¢ edge that matters at scale. When screening arb, carry two decimal places in probability space, then map back to cents for execution.
Habit
Keep a notes field: format received → raw P each leg → sum → devigged P → Kalshi ask → gap. Five arrows, one screen, fewer sign errors on American favorites.
What comes next
Next: expected value—turning probability disagreements into dollars per share and deciding when positive EV is worth the click after fees.