Published July 14, 2026. This is a reproducible mathematical study, not a record of actual wagers or a claim about future returns.
Quick Answer
Odds shopping can change a bet from negative expected value to positive expected value without changing the underlying pick. At a 55% true win probability, a $100 wager has an expected profit of $5.00 at -110, $7.38 at -105, and $10.00 at +100. A numerical prop-line change can matter even more, but its value depends on the outcome distribution rather than a universal conversion rate.
The important distinction is price versus line. American odds determine payout and break-even probability directly. A player-prop line such as 24.5 or 25.5 changes the probability of winning; how much it changes that probability depends on the player’s projection, volatility, market rules, and the distribution of possible outcomes. This study quantifies both effects while keeping those assumptions visible.
Study Design
This analysis uses two controlled experiments. Experiment one holds the event’s true win probability fixed and changes only the American price. Experiment two holds the price at -110 and changes only a hypothetical player-prop line. No sportsbook feed, proprietary prediction, selected winner, or historical result is used.
- Price experiment: compare American odds from -130 through +130 at fixed win probabilities of 52.5%, 55%, and 60%.
- Line experiment: model a hypothetical Over with a projection of 25.0 units and a normal outcome distribution with a 5.0-unit standard deviation.
- Sensitivity test: repeat the 24.5-versus-25.5 comparison across standard deviations from 3 to 8 units.
- Unit: expected profit per $100 staked. A positive number is mathematical positive expected value under the assumption; it is not a promised return.
Table 1: Price Shopping at a 55% Win Probability
Break-even probability is the win rate required for an offered price to have zero expected profit before considering pushes, voids, or other settlement rules. The same 55% opinion can be unplayable at one price and attractive at another.
| American odds | Break-even probability | Expected profit per $100 at 55% | Expected profit per $100 at 52.5% | Expected profit per $100 at 60% |
|---|---|---|---|---|
| -130 | 56.52% | -$2.69 | -$7.12 | +$6.15 |
| -125 | 55.56% | -$1.00 | -$5.50 | +$8.00 |
| -120 | 54.55% | +$0.83 | -$3.75 | +$10.00 |
| -115 | 53.49% | +$2.83 | -$1.85 | +$12.17 |
| -110 | 52.38% | +$5.00 | +$0.23 | +$14.55 |
| -105 | 51.22% | +$7.38 | +$2.50 | +$17.14 |
| +100 | 50.00% | +$10.00 | +$5.00 | +$20.00 |
| +105 | 48.78% | +$12.75 | +$7.62 | +$23.00 |
| +110 | 47.62% | +$15.50 | +$10.25 | +$26.00 |
| +120 | 45.45% | +$21.00 | +$15.50 | +$32.00 |
| +130 | 43.48% | +$26.50 | +$20.75 | +$38.00 |
The 55% column shows the direct value of price shopping. A bettor with the exact same probability estimate receives a different expected return because a better price pays more on a win and requires a lower hit rate to break even. The probability estimate still has to be honest. If the real chance is below the assumed 55%, the apparent edge can shrink or disappear.
Table 2: A One-Unit Prop-Line Difference
This experiment uses a hypothetical player outcome centered at 25.0 with a 5.0-unit standard deviation. All offers are half-point Overs priced at -110, so pushes are excluded in the illustration. The normal distribution is a teaching assumption, not a claim that every player or stat follows it.
| Hypothetical Over line | Modeled win probability | Expected profit per $100 at -110 |
|---|---|---|
| Over 22.5 | 69.15% | +$32.01 |
| Over 23.5 | 61.79% | +$17.96 |
| Over 24.5 | 53.98% | +$3.06 |
| Over 25.5 | 46.02% | -$12.15 |
| Over 26.5 | 38.21% | -$27.06 |
| Over 27.5 | 30.85% | -$41.10 |
Under these assumptions, Over 24.5 and Over 25.5 are not close substitutes. The one-unit move crosses the center of the modeled distribution. The expected-profit difference is $15.21 per $100 staked even though both bets carry -110 odds. That does not mean every one-unit move is worth $15.21. It means line value depends on where the line sits relative to the distribution.
Table 3: Why Volatility Changes the Value of a Line
A tighter distribution places more probability near the projection, so a one-unit line change around the center matters more. A wider distribution spreads outcomes farther away, reducing the probability mass crossed by the same line move.
| Assumed standard deviation | Over 24.5 win probability | Over 24.5 EV | Over 25.5 win probability | Over 25.5 EV | EV difference |
|---|---|---|---|---|---|
| 3 units | 56.62% | +$8.09 | 43.38% | -$17.18 | $25.27 |
| 4 units | 54.97% | +$4.95 | 45.03% | -$14.04 | $18.99 |
| 5 units | 53.98% | +$3.06 | 46.02% | -$12.15 | $15.21 |
| 6 units | 53.32% | +$1.79 | 46.68% | -$10.88 | $12.68 |
| 7 units | 52.85% | +$0.89 | 47.15% | -$9.98 | $10.87 |
| 8 units | 52.49% | +$0.21 | 47.51% | -$9.30 | $9.51 |
This sensitivity table is why generic statements such as “a half point is worth X%” should be treated cautiously for player props. The value changes by stat type, player role, distribution shape, integer versus half-point settlement, and the location of the line. A price can be compared universally through payout math; a numerical line requires a model of outcomes.
Reproducible Formulas
For positive American odds A, decimal odds equal 1 + A/100 and break-even probability equals 100/(A + 100). For negative American odds -A, decimal odds equal 1 + 100/A and break-even probability equals A/(A + 100).
Expected profit per dollar staked is:
EV = p × (decimal odds – 1) – (1 – p)
Multiply by 100 to express the result per $100 staked. In the prop-line experiment, the win probability is the area above the offered line under a normal distribution with mean 25.0 and the stated standard deviation. Readers can reproduce every displayed value with those formulas; rounding occurs only for presentation.
Why a Fair Projection Is Not Automatically a Fair Bet
A projection describes the center or expected value of an outcome, while a wager resolves against a specific line at a specific payout. Two distributions can share the same 25.0 projection and assign very different probabilities to Over 24.5 because their volatility, skew, and tail behavior differ. Even a well-calibrated win probability must then be compared with the offered price. This is why the study reports the assumed distribution, modeled win probability, and payout math separately instead of treating a projection gap as a complete betting edge.
What the Study Does and Does Not Prove
- It proves the arithmetic relationship among price, break-even probability, assumed win probability, and expected value.
- It demonstrates why a better numerical prop line can be more important than a small price improvement.
- It does not prove that the hypothetical 25.0 projection or normal distribution is correct for a real player.
- It does not remove sportsbook margin, estimate correlation, model limits, or account for void and push rules.
- It does not guarantee profit. A bad probability estimate can make a mathematically attractive price misleading.
Practical Odds-Shopping Checklist
- Confirm that the player, market, period, direction, and settlement rules match.
- Compare the numerical line before comparing the price. Over 24.5 and Over 25.5 are different bets.
- If the lines match, choose the better payout after checking any meaningful rules difference.
- Record the exact line and odds available when the decision was made.
- Recalculate expected value using a probability estimate that includes uncertainty.
- Pass when the line moved beyond the modeled edge. A stale opinion is not a current price.
Use the free two-offer player-prop comparison to compare manually entered offers, the lite prop-bet analyzer for implied probability and expected-value arithmetic, and the odds-shopping guide for the broader workflow. Those public tools intentionally do not pull live sportsbook inventory or reveal PropsBot’s full model.
Frequently Asked Questions
Is -105 always better than -110?
Yes, when the event, side, numerical line, limits, and settlement rules are identical. The -105 price requires a lower break-even win rate and pays more profit for the same stake.
Is a better line always more important than a better price?
Not always. A numerical line changes the chance of winning, while price changes payout. The relative value depends on how much probability lies between the two lines and how large the price difference is.
Why not assign one universal value to a half point?
Because outcome distributions differ. A half point near a high-probability cluster can be valuable, while the same move in a sparse tail can be worth much less.
Does positive expected value mean the next bet should win?
No. Expected value is a long-run average under a probability assumption. Any individual bet can lose, and the assumption itself can be wrong.
Is this study based on PropsBot picks?
No. It is a transparent mathematical illustration. PropsBot’s dated performance evidence is separately documented in the MLB model performance report and public results ledger.
