What Sunday Can Do: Thirteen Games as One Distribution

Sunday as One Object

Thirteen of the sixteen week-1 games kick on Sunday, September 13, and every number on this page is a price written before any of them has. The ledger's thirteen probabilities have been on the record since August 12. Other pages here have walked the sixteen rows, sorted them by conviction and priced them against the June lines. This one does something narrower: it treats Sunday as a single object — thirteen coins of known weight tossed at once — and asks what that object can do to the three things this site keeps score with. The week's miss budget. The season's. And the top ten of the board, which four games on Sunday can rearrange and one cannot touch.

The arithmetic is a convolution, not a simulation. Given thirteen independent probabilities the exact distribution of how many picks land has fourteen entries and takes one loop to compute, and I have computed it twice: once from the model's frozen numbers, and once from the market's vig-free numbers as of Friday morning's pull, which is the market's grade of the model's board before the games grade it themselves. The two distributions have the same mode and differ by a fifth of a game in expectation. They differ a great deal more on three individual games, and those three are where Sunday will actually be decided for the ledger.

The short version: the model expects 8.08 of its thirteen Sunday picks to land, gives eight exactly the single highest probability at 22.6%, and gives a losing Sunday — six or fewer — a 17.9% chance. Nine or more, the count that would leave the season budget under its opening 99.11, is 41.1%. The market, asked the same question about the model's picks, says 7.86, 21.0% and 36.0%. And the number one spot on the board is not in play in any of the 8,192 ways Sunday can go.

The Distribution

The thirteen pick probabilities run from .5096 (the Giants over Dallas, the coin flip on the board) to .7534 (Jacksonville over Cleveland). Their sum is 8.0753, which is the expected number of correct picks, and the standard deviation of the count is 1.72 games — narrower than thirteen fair coins' 1.80, because a coin weighted away from a half has less variance, not more. The exact probabilities:

Seven to ten correct covers 74.6% of the model's distribution. An all-chalk Sunday is one in 547. A losing Sunday is 17.9%, which is to say the model expects to post a losing Sunday roughly one week in six even if every one of its probabilities is exactly right; a page that greets the first 6–7 with alarm has not read the table. The market's column is a little heavier on the left, for a reason the third section explains, but both columns put their mass in the same place. Whatever Sunday does, it will almost certainly do it with seven, eight, nine or ten.

The Exhibit

What Each Count Does to the Budget

The miss budget opened the season at 99.11 expected wrong picks. Two games have graded: Wednesday's carried .3215 expected misses and landed, Thursday's carried .4211 and did not, so the two together were expected to produce .7426 misses and produced one, and the season stands at 99.37 this morning. Sunday's thirteen carry 4.92 more; through Sunday the model expected 5.67 misses in fifteen games. The rightmost column of the table above is what the season's expectation will read on Sunday night for each count: 99.11, less the 5.67 expected from the fifteen graded games, plus the misses actually recorded.

The break-even is nine. At eight correct the season reads 99.44, a third of a game over its opening number; at nine it reads 98.44 and the ledger is, for the first time, ahead of its own August self. The model gives nine or more 41.1%. Each pick above or below the mode moves the season by exactly one game, because a miss is a miss whatever the probability was — the .4211 that Thursday's game carried is gone from the expected column and the 1 it produced sits in the realised one, and the same will be true of every row on Sunday. The budget frame is unforgiving in that specific way: it does not grade on a curve, it grades on the count.

For the week, the picture is the one the learning-curve page pre-registered with one adjustment. That page said 7 to 13 correct covered 93.6% of the sixteen-game distribution. With one hit and one miss banked, the week now expects 9.66 correct against the original 9.91, and the same 7–13 window covers 94.9%: the expectation fell and the window got safer, because two of the sixteen coins have stopped spinning and the distribution over the rest is narrower. Monday's game, Denver at .5805 in Kansas City, carries the last .42 of the week's budget and is priced on its own page.

The Market's Grade of the Same Board

Take the thirteen moneyline pairs in the game file as of Friday morning's pull, strip the hold — between 4.2% and 4.6% on every game — and you have the market's probability for each of the model's picks. Run the same convolution. The market expects 7.86 of the model's thirteen picks to land, against the model's 8.08, a fifth of a game apart; puts the mode at eight (23.0%); gives a losing Sunday 21.0% against the model's 17.9%; and gives nine or more 36.0% against 41.1%. The market's own favourites, by the market's own numbers, expect 8.29 correct, so the market thinks its board is 0.43 of a game better than the model's this Sunday. That is the whole disagreement in one number, and it is small.

It is small because ten of the thirteen games are agreed on. The market's favourite is the model's pick in ten; the mean absolute disagreement across all thirteen is 6.1 points of probability; and on Detroit over New Orleans the two are within a point (.7343 against .7258). The three games where the market sides against the model's pick are the three that matter:

Miami at Las Vegas is the argument of the week and has been since June, when the road-favourites page measured it at 19.5 points; against Friday's pull it is 18.9, so the summer changed almost nothing. The model has Miami as a road favourite because its rating has the Raiders thirty-first on the board at 1351.0 and the Dolphins 108 points better; the market has Las Vegas by three at home. One of them is wrong by a lot, and Sunday will not say which — a .5857 pick losing is a two-in-five event, and a .6038 favourite losing is very nearly the same event from the other side. The Giants game is the second split and it is a different kind: the model is at a coin flip and the market has moved the other way, from Dallas by 1.5 in June to Dallas by 3 now. Cincinnati is the one game where the market is the more confident side by a wide margin, .6369 to .5573, which is the model being quieter than the price rather than louder.

The three splits are also, exactly, where the market's column in the first table gets its extra weight on the left. Under the market's numbers the model's picks in those three games land 1.29 times between them; under the model's, 1.65. The other ten games contribute almost identically under either set of beliefs. Sunday's grade for this ledger will be written in Las Vegas, the Meadowlands and Houston, and everything else is the model and the market agreeing to be wrong or right together.

What the Market Learned Over the Summer

The site's bundled game file is a June snapshot, and the refresh that graded Thursday's game also brought down Friday morning's lines for the fourteen games still to play, so the two can be laid side by side. Between June and September 11 exactly one favourite changed: Green Bay was −1.5 at Minnesota in June and Minnesota is −1.5 now, which moves the market onto the model's side of that game. The biggest move in probability was in Nashville, where Tennessee went from a three-point home favourite (.6038 vig-free) to a 1.5-point one (.5325), 7.1 points toward the Jets; Pittsburgh went the other way, from 3 to 5.5 over Atlanta, 6.0 points; Chicago from −2.5 to −3 at Carolina; Dallas from −1.5 to −3 at the Giants; Jacksonville from 7.5 to 8.5. The week's longest line shortened, the Chargers from 11.5 to 9.5 over Arizona.

Scored against the model's picks, the market moved toward the model in seven games (Pittsburgh, Minnesota, Chicago, Jacksonville, Philadelphia, Cincinnati and, by half a point, Miami) and away in six (Tennessee, the Giants, the Chargers, Baltimore, Houston, Detroit). The net is 5.3 points of pick probability across thirteen games, or 0.05 expected correct: in June the market gave the model's Sunday picks 7.81; now 7.86. A summer of camps, cuts and quarterback news moved the market's opinion of this model's board by one-twentieth of a game. Whatever the market learned between June and September, almost none of it bore on the question of which side the frozen ratings had right.

The two games that have already been played are a caution on reading any of this as a lean. On Wednesday the market moved toward New England between June and the close, and Seattle won; on Thursday it moved toward the Rams between the morning pull and the close, and San Francisco won. Two games. The September 11 numbers here are a snapshot two days before kickoff, not closes, and the closes will differ.

What Sunday Can Do to the Top Ten

Five of the board's top ten play on Sunday: Buffalo (#2), Houston (#4), Philadelphia (#7), Jacksonville (#9) and Detroit (#10). Denver (#3) plays Monday. The other four — Seattle, San Francisco, New England, the Rams — have played already and sit out. Every price below applies the ledger's own update rule to the current ratings at stated margins, both ways.

Buffalo at Houston is the only game between two top-ten teams, 1615.9 against 1605.6, 10.3 points apart, Houston at .554 with the home edge. It is also the only game on the slate with a result that changes nothing: a one-point Buffalo win leaves Buffalo second and Houston fourth, because Buffalo gains 7.55 and Houston's 1598.1 still clears San Francisco's 1593.8. Every other result moves seats. A one-point Houston win makes Houston third and Buffalo fourth; Houston by seven (+18.23) is Houston second and Buffalo fourth; Buffalo by seven (Houston −22.65) is Buffalo second and Houston sixth; Buffalo by fourteen drops Houston to eighth. Across the whole of Sunday at a touchdown everywhere, Houston finishes second with probability .554, seventh with .325 and sixth with .121, and Buffalo is second (.446) or fourth (.554) — one game decides it. No Buffalo margin reaches Seattle: the most Buffalo can gain at the file's record margin is 44.6 points against a 67.1-point gap. Seattle is first on Sunday night in every one of the 8,192 outcomes.

The biggest swings available are all upsets of the four longest favourites, because the engine pays out at the odds it quoted. Cleveland beating Jacksonville by seven moves 28.79 rating points and drops the Jaguars from ninth to twelfth; Jacksonville winning by the same seven moves 9.43 and leaves them ninth, and the ratio between the two, 3.05, is the prior odds on the pick. Arizona over the Chargers by seven is 28.51; New Orleans over Detroit is 28.27 and puts the Lions thirteenth; Washington over Philadelphia is 28.09 and puts the Eagles twelfth — and that last result, alone, reorders twelve rows of the board, the busiest single outcome on the slate, because the Eagles would fall through the tightest stretch of the board where six teams sit within 25 points. A fourteen-point Eagles win lifts them to fifth. Monday, any Denver win makes the Broncos second; a seven-point Kansas City win drops them to fifth.

How much of the top ten changes hands. Enumerate all 8,192 Sunday outcomes at a touchdown margin, weight each by the model's own probabilities, and count the seats among today's top ten whose occupant changes: never fewer than three, never more than seven. Four seats is the single most likely count (28.8%), then six (25.0%), then three (18.0%), five (16.1%) and seven (12.1%). Across the whole 32-row board the expected number of ranks that change on Sunday is 22.4, never fewer than fourteen and never more than 29. The rankings history put a full sixteen-game week 1 at 26.2 ranks changed on average, so thirteen games at a fixed margin land where thirteen-sixteenths of that history would suggest. The rankings page will draw none of this as movement until Monday night's game grades and the week's snapshot freezes; the arrows appear Tuesday.

What This Page Does Not Show

Every game on this page is unplayed. The distribution, the budget table, the market comparison and the board consequences are prices from the frozen ledger and the current ratings as of September 11. Nothing here is a result.

Independence is assumed. The convolution treats thirteen games as independent coins. They are played by different teams on the same afternoon and the assumption is close to right, but a slate-wide effect — weather across a region, an officiating emphasis — would widen the true distribution slightly beyond the one shown.

The market numbers are a snapshot. They are the game file's moneylines as pulled on the morning of September 11, two days before kickoff, with the hold stripped proportionally. They are not closing prices, and the closing prices are the ones a market comparison should be scored against once the games are final.

The board consequences fix the margin. Seven points everywhere is a stated assumption that makes the enumeration tractable; real margins vary, and a blowout in one game and a field goal in another would produce a different set of seat changes with the same thirteen winners.

The budget frame counts picks, not probabilities. A miss at .5096 and a miss at .7534 are both one miss in the season column. That is the point of the frame, and also its limit; the Brier column on the ledger page is where the probabilities are graded.

No rosters, no injuries, no quarterbacks. The model's thirteen numbers are 2025 results plus a third of a regression. The market's numbers know everything that happened this summer. Where they disagree, that is usually why.

Method and Sources

Two files and one function. The frozen ledger at /data/predictions.json supplies the thirteen probabilities and the current ratings; the nflverse game log at /data/games.csv supplies the lines, in its June form (the bundle the site serves) and its September 11 form (saved by the companion harness as a dated snapshot, because the build republishes the June bundle over the served copy); explainer_src/nfl_elo.py supplies the update rule for the board consequences. The harness is explainer_src/make_what_sunday_can_do_chart.py. The distribution is the exact Poisson-binomial, one loop:

def pmf(ps):                      # ps = the 13 pick probabilities, max(p_home, 1 - p_home)
    d = [1.0]
    for p in ps:
        nd = [0.0] * (len(d) + 1)
        for k, v in enumerate(d):
            nd[k]     += v * (1 - p)   # this pick misses
            nd[k + 1] += v * p         # this pick lands
        d = nd
    return d                      # d[k] = P(exactly k of 13 land); mode 8 (.2262), P(k <= 6) = .1792

The script asserts the ledger's state (two graded, fourteen not, every remaining row an August 12 lock), the thirteen probabilities and their sum, every entry of both distributions quoted above, the standard deviation, the week's conditional expectation and window, the miss-budget arithmetic for every count and the break-even at nine, the market's vig-free numbers and holds for all thirteen games as of the pull and in June, the agreement count, the three splits by name and size, the summer moves game by game and the one favourite flip, every board consequence stated (Buffalo at Houston at 1, 7 and 14 both ways, the four upset swings, the Eagles' twelve rows, Denver on Monday), the full 8,192-outcome enumeration, and that Seattle is first in all of them: 129 assertions, all green as of September 11, 2026.

Sources: this site's frozen ledger and power-rankings generator, both published in full; the nflverse public game log (games.csv), bundled at /data/games.csv in its June form, with the September 11 lines quoted from that morning's pull as a snapshot. The Poisson-binomial distribution is standard; the convolution above is its definition.

Further reading

• The First Miss — the companion page: San Francisco 27, Rams 7, graded against its own pre-registrations
• The Opener, Graded — Wednesday's hit, and why one game grades a pick and not a probability
• Nobody Moved — the rankings history: 26.2 of 32 ranks change in a full week 1
• The Prediction Ledger — the thirteen frozen numbers, one for two so far
• Power Rankings — the board Sunday will rearrange, with the movement column that fills in Tuesday
• The Miss Budget — the 99.11 this page's rightmost column is measured against
• The Model Gets Louder, Not Smarter — the 7–13 window, pre-registered before the week
• Week 1 Predictions, By the Numbers — the sixteen-game version of the same convolution, from August
• Nine Games on One Shelf — why so many of these thirteen sit between .55 and .60
• What a 60% Pick Is Worth — the same board against the June lines, and the audit that says the market is usually right
• The Road Favorite Discount That Isn't There — Miami at Las Vegas, the argument of the week since June
• Kansas City Is 17th — Monday's game, priced in advance
• The Week 10 Problem That Isn't — the pre-registered bars a whole week has to clear to be surprising