{"id":3120,"date":"2026-07-20T10:00:00","date_gmt":"2026-07-20T10:00:00","guid":{"rendered":"https:\/\/oddspapi.io\/blog\/?p=3120"},"modified":"2026-06-22T13:57:18","modified_gmt":"2026-06-22T13:57:18","slug":"poisson-soccer-model-python","status":"publish","type":"post","link":"https:\/\/oddspapi.io\/blog\/poisson-soccer-model-python\/","title":{"rendered":"Poisson Soccer Model in Python: Reconstruct Correct-Score Odds from the Market"},"content":{"rendered":"

You can see the moneyline. You can see the over\/under 2.5. But what’s the market’s implied probability of a 2–1? Or a 0–0? Or the chance of exactly 4 goals? Books quote a handful of headline lines and leave the rest of the score distribution unstated — even though it’s sitting right there, implied by the prices they do<\/em> publish.<\/p>\n

This guide shows you how to recover it. We’ll take a sharp book’s de-vigged 1X2 and over\/under prices from the OddsPapi odds API<\/a>, fit a Poisson model<\/strong> in pure Python (no libraries beyond the standard math<\/code> module), and reconstruct the entire correct-score matrix — every scoreline, every alternative total, BTTS, all of it. Then we’ll be honest about where the basic Poisson model breaks, and what the pros do about it.<\/p>\n

Why Reconstruct Instead of Predict?<\/h2>\n

There are two ways to get a score distribution. Most tutorials build a predictive<\/em> model: scrape years of match results, fit attack\/defence ratings, project goals. That needs a results-and-stats database you probably don’t have, and it’s only as good as your feature engineering.<\/p>\n

The other way — the one sharps actually use as a baseline — is to reconstruct the distribution the market has already priced<\/strong>. Pinnacle moves billions in handle and its closing line is the most accurate public estimate of a match’s outcome. Its 1X2 and over\/under prices already encode a full goal-expectancy view. We just invert it. No results database, no training, no overfitting — the sharpest forecast in the world, decoded into every market the book didn’t bother to quote.<\/p>\n

\n\n\n\n\n\n\n\n\n
The Old Way<\/th>\nOddsPapi + Poisson<\/th>\n<\/tr>\n<\/thead>\n
Scrape correct-score odds book by book<\/td>\nOne \/odds<\/code> call returns the lines you need<\/td>\n<\/tr>\n
Build a predictive model from a results DB you don’t have<\/td>\nCalibrate from the market’s own closing line<\/td>\n<\/tr>\n
Get only the markets the book chooses to quote<\/td>\nDerive every<\/em> scoreline and alternative total yourself<\/td>\n<\/tr>\n
One bookmaker’s view<\/td>\nDe-vig the sharpest of 350+ books, or blend a consensus<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/figure>\n

The Model in 60 Seconds<\/h2>\n

Goals in a football match are well approximated by a Poisson process<\/strong>: each team scores at some average rate (lambda, λ) over 90 minutes, and the number of goals follows a Poisson distribution. If we know the home team’s expected goals λhome<\/sub> and the away team’s λaway<\/sub>, the probability of any exact scoreline i–j<\/em> is just the product of two Poisson probabilities:<\/p>\n

P(score = i-j) = poisson(i, lambda_home) * poisson(j, lambda_away)<\/code><\/pre>\n
So the whole job reduces to two numbers: λhome<\/sub> and λaway<\/sub>. We back them out of the market in two steps:<\/p>\n
    \n
  1. Total goals<\/strong> — the de-vigged over\/under 2.5 price tells us λhome<\/sub> + λaway<\/sub>.<\/li>\n
  2. The split<\/strong> — the de-vigged 1X2 tells us how to divide that total between the two teams.<\/li>\n<\/ol>\n
    Step 1: Authenticate and Pull the Lines<\/h2>\n
    OddsPapi auth is a query parameter, not a header. Grab a free API key<\/a> and pull the odds for one fixture. We only need a sharp book here — Pinnacle — but the endpoint returns all 350+ books on the fixture, so you can swap in a consensus later.<\/p>\n
    import requests, time\nfrom math import exp, factorial\n\nAPI_KEY = \"YOUR_API_KEY\"\nBASE = \"https:\/\/api.oddspapi.io\/v4\"\n\ndef get(path, **params):\n    # GET with the ~0.9s same-endpoint cooldown handled\n    params[\"apiKey\"] = API_KEY\n    for _ in range(6):\n        r = requests.get(f\"{BASE}\/{path}\", params=params)\n        try:\n            data = r.json()\n            if data:\n                return data\n        except ValueError:\n            pass\n        time.sleep(1.0)\n    return {}\n\n# A World Cup 2026 fixture: Argentina v Austria (snapshot, June 22 2026)\nfixture_id = \"id1000001666457022\"\nodds = get(\"odds\", fixtureId=fixture_id)\nmarkets = odds[\"bookmakerOdds\"][\"pinnacle\"][\"markets\"]\n\ndef price(market_id, outcome_id):\n    try:\n        return markets[str(market_id)][\"outcomes\"][str(outcome_id)][\"players\"][\"0\"][\"price\"]\n    except (KeyError, TypeError):\n        return None\n\n# Full Time Result 1X2 (market 101): Home=101, Draw=102, Away=103\nh, d, a = price(101, 101), price(101, 102), price(101, 103)\n# Over\/Under 2.5 goals (market 1010): Over=1010, Under=1011\nover25, under25 = price(1010, 1010), price(1010, 1011)\nprint(\"1X2 :\", h, d, a)          # 1.476 4.5 8.0\nprint(\"O\/U2.5:\", over25, under25) # 2.01 1.9\n<\/code><\/pre>\n
    Decimal odds carry the bookmaker’s margin (the “vig”). Before we can read them as probabilities we have to strip it out.<\/p>\n
    Step 2: De-vig to Fair Probabilities<\/h2>\n
    The simplest de-vig is proportional: take the inverse of each price (the raw implied probability), then normalise so they sum to 1. For a deeper treatment of the power and Shin methods, see our no-vig odds guide<\/a> — proportional is fine for this walkthrough.<\/p>\n
    def devig(*odds):\n    inv = [1 \/ o for o in odds]\n    total = sum(inv)\n    return [i \/ total for i in inv]\n\np_home, p_draw, p_away = devig(h, d, a)\np_over, p_under = devig(over25, under25)\n\nprint(f\"Fair 1X2 : H={p_home:.3f} D={p_draw:.3f} A={p_away:.3f}\")\n# Fair 1X2 : H=0.661 D=0.217 A=0.122\nprint(f\"Fair O\/U : Over={p_over:.3f} Under={p_under:.3f}\")\n# Fair O\/U : Over=0.486 Under=0.514\n<\/code><\/pre>\n
    So the market makes Argentina a 66.1% favourite and sees the game as a near coin-flip on 2.5 goals. Those two facts are enough to pin the entire score distribution.<\/p>\n
    Step 3: Solve for Total Goal Expectancy<\/h2>\n
    Under a Poisson with mean λtotal<\/sub>, the probability of 3 or more<\/strong> goals (i.e. “over 2.5”) has a closed form. We don’t even need to invert it analytically — a few rounds of bisection nail λtotal<\/sub> to the de-vigged over price.<\/p>\n
    def poisson(k, lam):\n    return exp(-lam) * lam ** k \/ factorial(k)\n\ndef prob_over(line, lam):\n    # P(total goals > line) for a .5 line, e.g. over 2.5 -> P(N >= 3)\n    threshold = int(line + 0.5)\n    return 1 - sum(poisson(k, lam) for k in range(threshold))\n\ndef solve_lambda_total(p_over, line=2.5):\n    lo, hi = 0.1, 6.0\n    for _ in range(60):\n        mid = (lo + hi) \/ 2\n        if prob_over(line, mid) < p_over:\n            lo = mid\n        else:\n            hi = mid\n    return (lo + hi) \/ 2\n\nlam_total = solve_lambda_total(p_over)\nprint(f\"lambda_total = {lam_total:.3f}\")   # 2.617\n<\/code><\/pre>\n
    The market is pricing in about 2.62 total goals<\/strong>. Now we split it.<\/p>\n
    Step 4: Split the Total Using the 1X2<\/h2>\n
    Given a fixed total, the home\/away split is a single degree of freedom: the bigger Argentina's share, the higher its win probability. We bisect on that share until the model's home-win probability matches the de-vigged 1X2.<\/p>\n
    def outcome_probs(lam_h, lam_a, max_goals=12):\n    # Home-win \/ draw \/ away-win from an independent Poisson grid\n    ph = pd = pa = 0.0\n    for i in range(max_goals):\n        for j in range(max_goals):\n            p = poisson(i, lam_h) * poisson(j, lam_a)\n            if i > j:   ph += p\n            elif i == j: pd += p\n            else:        pa += p\n    return ph, pd, pa\n\ndef split_lambda(lam_total, p_home_target):\n    lo, hi = 0.50, 0.95   # home share of total goals\n    for _ in range(60):\n        share = (lo + hi) \/ 2\n        lam_h = share * lam_total\n        ph, _, _ = outcome_probs(lam_h, lam_total - lam_h)\n        if ph < p_home_target:\n            lo = share\n        else:\n            hi = share\n    share = (lo + hi) \/ 2\n    return share * lam_total, lam_total - share * lam_total\n\nlam_home, lam_away = split_lambda(lam_total, p_home)\nprint(f\"lambda_home = {lam_home:.3f}  lambda_away = {lam_away:.3f}\")\n# lambda_home = 1.914  lambda_away = 0.703\n<\/code><\/pre>\n
    Argentina is priced to score ~1.91, Austria ~0.70. That's our complete model — two numbers that came entirely from the market.<\/p>\n
    Step 5: Reconstruct Every Market<\/h2>\n
    With λhome<\/sub> and λaway<\/sub> in hand, build the score matrix once and read any market straight off it — including ones Pinnacle never quoted.<\/p>\n
    MAXG = 12\nmatrix = [[poisson(i, lam_home) * poisson(j, lam_away)\n           for j in range(MAXG)] for i in range(MAXG)]\n\n# Top correct scores\nscores = [((i, j), matrix[i][j]) for i in range(6) for j in range(6)]\nscores.sort(key=lambda x: -x[1])\nprint(\"Most likely scorelines:\")\nfor (i, j), p in scores[:6]:\n    print(f\"  {i}-{j}: {p*100:4.1f}%   fair odds {1\/p:5.1f}\")\n\n# Derived markets\nbtts = sum(matrix[i][j] for i in range(1, MAXG) for j in range(1, MAXG))\nover_15 = sum(matrix[i][j] for i in range(MAXG) for j in range(MAXG) if i + j >= 2)\nover_35 = sum(matrix[i][j] for i in range(MAXG) for j in range(MAXG) if i + j >= 4)\nprint(f\"BTTS yes : {btts*100:.1f}%   (fair odds {1\/btts:.2f})\")\nprint(f\"Over 1.5 : {over_15*100:.1f}%\")\nprint(f\"Over 3.5 : {over_35*100:.1f}%\")\n<\/code><\/pre>\n
    Output:<\/p>\n
    Most likely scorelines:\n  1-0: 14.0%   fair odds   7.2\n  2-0: 13.4%   fair odds   7.5\n  1-1:  9.8%   fair odds  10.2\n  2-1:  9.4%   fair odds  10.6\n  3-0:  8.5%   fair odds  11.7\n  0-0:  7.3%   fair odds  13.7\nBTTS yes : 43.1%   (fair odds 2.32)\nOver 1.5 : 80.4%\nOver 3.5 : 22.0%<\/code><\/pre>\n
    You now have a fair price for every correct score, every alternative total, and BTTS — reconstructed from two market inputs. Cross-check the derived over\/under 3.5 and 1.5 against the book's actual prices on the same fixture and they line up almost exactly, which is the point: the model isn't guessing, it's decoding.<\/p>\n
    Where Basic Poisson Breaks (Read This Before You Bet)<\/h2>\n
    Here's the honest part most tutorials skip. The independent Poisson model has a known, systematic flaw: it assumes the two teams' goals are independent<\/strong>. Real matches don't work that way — game state, red cards, and chasing a result all correlate the two scorelines. Compare our reconstruction to the book's actual both-teams-to-score line:<\/p>\n
    \n\n\n\n\n\n\n\n\n
    Market<\/th>\nPoisson model<\/th>\nPinnacle (de-vigged)<\/th>\nGap<\/th>\n<\/tr>\n<\/thead>\n
    Over 2.5<\/td>\n48.6%<\/td>\n48.6%<\/td>\n0.0<\/td>\n<\/tr>\n
    Home win<\/td>\n66.1%<\/td>\n66.1%<\/td>\n0.0<\/td>\n<\/tr>\n
    Draw<\/td>\n21.0%<\/td>\n21.7%<\/td>\n-0.7<\/td>\n<\/tr>\n
    BTTS yes<\/strong><\/td>\n43.1%<\/strong><\/td>\n47.6%<\/strong><\/td>\n-4.5<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/figure>\n
    The markets we calibrated against match perfectly (they have to). But BTTS is 4.5 points too low<\/strong>, and the draw is slightly under too. That's not noise — independent Poisson always<\/em> under-prices both-teams-to-score and draws, because in reality low-scoring outcomes cluster. This is exactly why pros reach for the Dixon–Coles<\/strong> adjustment, which adds a correlation term (ρ) to inflate the 0–0, 1–0, 0–1 and 1–1 cells. Our same-game parlay correlation guide<\/a> digs into why multiplying \"independent\" leg probabilities fails for the same reason.<\/p>\n
    Treat the basic model as a fast, transparent baseline — great for filling in unquoted scorelines — not as a value-detection engine. If your reconstructed price beats a soft book, the gap is at least as likely to be the model's independence error as it is a real edge.<\/p>\n
    From One Book to a Consensus — and Back in Time<\/h2>\n
    Two upgrades make this genuinely useful:<\/p>\n