Six Of Seven Legs Load On The Same Variable
Published August 24, 2026 | Shared Factor Model | MLB Prediction Data Lab
A seven leg card is normally summarised by one number, the weighted break even, and that number here is 56.32 percent. It is a correct number and it is an incomplete one, because it describes a card of seven independent bets and this is not one. Six of the seven legs are driven in part by the same latent variable: how much scoring happens in Major League Baseball on the night of August 24, 2026. This model quantifies what that does. The answer is that it does nothing at all to expected value and quite a lot to everything else.
The Card And The Loadings
Each leg is written below with its posted price, its stake, its break even and the sign of its loading on the shared factor. A positive loading means the leg wins when scoring is suppressed.
| Leg | Price | Stake | Break even | Profit if won | Factor loading |
|---|---|---|---|---|---|
| Pirates and Padres under 7.5 | -115 | 2.5 | 53.49 percent | 2.174 | Run environment, positive |
| Angels team total under 3.5 | -140 | 2.0 | 58.33 percent | 1.429 | Run environment, positive |
| Mariners team total under 3.5 | -150 | 2.0 | 60.00 percent | 1.333 | Run environment, positive |
| Cleveland Guardians moneyline | -160 | 1.0 | 61.54 percent | 0.625 | Run environment, positive |
| Tigers and Rays under 6.5 | -110 | 1.0 | 52.38 percent | 0.909 | Run environment, positive |
| Twins and Athletics over 11 | -110 | 1.5 | 52.38 percent | 1.364 | Run environment, negative |
| Chicago Cubs moneyline | -138 | 1.5 | 57.98 percent | 1.087 | Independent |
| Card | — | 11.5 | 56.32 percent | 8.92 | — |
Five legs load positively, one loads negatively and one does not load at all. That single negative loading matters more than it looks, and it is dealt with below.
The Model
Let Z be a standard normal variable representing the night's run environment, high values meaning a high scoring night. For each leg i, draw a latent variable
Xi = √ρ · (−si Z) + √(1 − ρ) · εi
where si is the sign of the loading, εi is independent standard normal noise, and the leg is scored a winner when Xi falls below the normal quantile of its win probability. This is a one factor Gaussian copula. It preserves each leg's marginal win probability exactly, which is the whole point: the individual bets are unchanged, only their joint behaviour moves.
The marginals used are the break even probabilities themselves. That is a deliberate neutrality assumption. It says nothing about whether these are good bets, and it isolates the effect being measured. If the card carries an edge, every number below shifts up together and the shape conclusions are unaffected.
Independence First
Set rho to zero, run 200,000 trials, and the card behaves the way the single break even number implies.
| Statistic | rho = 0.00 |
|---|---|
| Mean profit and loss | −0.009 units |
| Standard deviation | 4.03 units |
| 5th percentile | −6.83 units |
| 95th percentile | +6.33 units |
| Five or more legs win | 34.8 percent |
| Two or fewer legs win | 13.4 percent |
| Losing night | 49.4 percent |
The mean is zero to within simulation noise, exactly as constructed.
Now Turn The Factor On
Set rho to 0.35, a moderate loading, and rerun.
| Statistic | rho = 0.00 | rho = 0.35 | Change |
|---|---|---|---|
| Mean profit and loss | −0.009 | −0.014 | None |
| Standard deviation | 4.03 | 4.58 | +13.7 percent |
| 5th percentile | −6.83 | −8.07 | −1.24 units |
| 95th percentile | +6.33 | +6.33 | None |
| Five or more legs win | 34.8 percent | 39.2 percent | +4.4 points |
| Two or fewer legs win | 13.4 percent | 18.7 percent | +5.3 points |
| Losing night | 49.4 percent | 48.1 percent | −1.3 points |
Read the first row before any of the others. The expected value did not move. Expectation is linear, so the mean of a sum is the sum of the means whatever the dependence structure. Anyone who tells you correlated legs are worse bets in expectation is describing something that is not true.
Read the rest and the picture is entirely different. The standard deviation rises 13.7 percent. Both tails get heavier at once, which is the signature of a common factor: good nights cluster and bad nights cluster. The fifth percentile, the number that actually matters for a bankroll, deteriorates by 1.24 units on an 11.5 unit card.
Why The 95th Percentile Did Not Move
Because of the negative loading. Twins and Athletics over 11 wins when scoring is high, which is precisely when the five unders and the Guardians moneyline are failing. On the best possible night for the five, that leg loses, and its 1.5 unit stake caps the top of the distribution regardless of rho. The upper tail is bounded by the card's own internal hedge while the lower tail is not.
That asymmetry is worth stating plainly: the over is not diversification in the useful direction. It trims the upside without meaningfully protecting the downside, because on a genuinely high scoring night one winning leg at 1.364 units does not offset six losers totalling 10 units of risk.
What Rho Is Actually Worth Here
Rho of 0.35 is an assumption, not a measurement. Nightly league run environment does move together across parks through weather, umpire zone and the general state of pitching staffs in late August, but no attempt is made here to estimate the number from data, and that is a real limitation of this page.
The direction of the result is robust to the value chosen. Any positive rho widens the distribution and leaves the mean alone. What changes with the value is the magnitude, and a reader who thinks 0.15 is more realistic should scale the 13.7 percent accordingly rather than dismiss the effect.
The Practical Reading
A card like this should be sized against its fifth percentile rather than its break even. Under independence the bad tail is 6.83 units. With a shared factor it is 8.07. If a bankroll is set so that a bad night costs no more than some fixed fraction, that fraction has been understated by roughly 18 percent by treating seven correlated legs as seven separate ones.
Nothing above says the card is wrong. It says the card is one bet with seven expressions, and it should be understood and staked as one.
Limitations
Marginal probabilities are set to the break even prices rather than to any independent estimate of true probability, so this page makes no claim about edge. Rho is assumed rather than fitted. The factor structure is one dimensional, when in reality park, weather and bullpen fatigue are separate influences. Two hundred thousand trials leaves roughly two hundredths of a unit of noise on the mean, which is why the mean is described as unchanged rather than as exactly equal.
Does correlation make the card a worse bet?
Not in expectation. It makes the card a riskier bet for the same expectation, which is a different and much more manageable problem.
Why is the Cubs moneyline treated as independent?
Because it is a side, not a total, and a side is close to indifferent to the overall run environment. A high scoring night in Phoenix helps whichever team hits more, not the favourite in particular.
How would this be improved?
By fitting rho on historical same night correlations between game unders and team total unders instead of assuming it, and by splitting the single factor into park, weather and umpire components.