Parlays, Probability, and Expected Value

Before any bet has expected value, two terms have to be precise. Both will appear in every example for the rest of the lesson.

Probability is how often something is expected to happen, expressed as a number between 0 and 1 (or 0 and 100 percent). A coin flip has a probability of 0.5 of landing heads. An NFL team that historically wins 60 percent of comparable matchups has a probability of 0.60 of winning the next one.

Odds are how a sportsbook prices a probability. Odds also encode the operator's margin (which we will name on Page 3). The dominant U.S. format is American odds, used by every legal U.S. sportsbook.

Reading American odds

American odds use a positive or negative number based on whether the bet is favored or not.

• Negative odds (e.g. -150, -200, -110) describe the favorite. The number says how much you must stake to win $100.
• -150 means: stake $150 to win $100.
• -200 means: stake $200 to win $100.
• Positive odds (e.g. +130, +250, +800) describe the underdog. The number says how much you win on a $100 stake.
• +130 means: stake $100 to win $130.
• +250 means: stake $100 to win $250.

You can scale up or down. A -110 line on a $11 stake wins $10. A +250 line on a $20 stake wins $50.

Implied probability

Every set of American odds maps to an implied probability - the probability the line price is implicitly charging for that outcome.

The formulas are simple enough to do on a whiteboard.

• Negative odds: implied probability = |odds| / (|odds| + 100).
• -110 -> 110 / (110 + 100) = 110 / 210 = 0.524 -> 52.4 percent.
• -150 -> 150 / 250 = 0.600 -> 60 percent.
• Positive odds: implied probability = 100 / (odds + 100).
• +130 -> 100 / 230 = 0.435 -> 43.5 percent.
• +250 -> 100 / 350 = 0.286 -> 28.6 percent.

If a coin flip were posted at "fair" odds, both sides would be priced at exactly +100 (which implies 50 percent). They are not. They are typically priced at -110 / -110 (which implies 52.4 percent on each side, totaling 104.8 percent). That extra 4.8 percent is the operator margin we will study on Page 3.

Worked translation: a real Saturday line

Take a generic NFL game. The sportsbook posts:

• Team A moneyline: -150
• Team B moneyline: +130

Translate to implied probability:

• Team A: 150 / 250 = 60.0 percent
• Team B: 100 / 230 = 43.5 percent

Sum: 103.5 percent. The total exceeds 100 percent because the operator built margin into both sides of the line. We will name and price that margin on Page 3.

A teaching note

Many students - and many adults - have an intuitive sense that "favorites win more often" without a precise translation. This page makes the translation precise. Once a student can convert -150 into 60 percent in their head, the rest of the unit becomes much easier, because every concept downstream (expected value, vig, parlay compounding) operates on probabilities, not on the raw odds notation.

If a student can do nothing else from Lesson 1 by the end, they should be able to translate American odds into implied probability without a calculator.