Monte Carlo Simulation: How to go broke eight times faster with 8 Eurojackpot lines

Every lottery player knows the saying: "One line is no line, you have to play a few more to boost your chances!" That's why the average player often fills out a complete ticket with 8 lines. Sounds like a solid strategy, right? Wrong. It’s actually the fastest way to systematically burn through your cash. To prove it, we wrote a Python simulation that uses historical payout data and cold, hard combinatorics to see who actually has any money left in their account at the end. What exactly is a Monte Carlo simulation? Named after the famous casino in Monaco, the Monte Carlo simulation is basically the brute-force approach to probability theory. Usually, mathematicians use a single, elegant formula to calculate the expected value. That formula will dryly inform you: "You lose an average of 1 Euro per Eurojackpot line." That might be mathematically correct, but emotionally, it's a bit of a snooze. It doesn't capture the true pain of slowly bleeding out financially. The Monte Carlo simulation throws that elegant formula right out the window. Its core concept is pure, raw computing power. Instead of just calculating the theoretical outcome, we let the computer simply play through reality thousands of times. It’s not an equation; it’s a simulation and iteration of real events. The computer spawns 1,000 fictional players. For every player and every draw over the last 10 years, it generates a random number based on the actual Eurojackpot probabilities. It simulates the real-world winning and (mostly) losing, step by step. At the end, we aren't looking at some abstract theoretical number, but at the very real, blood-red bank accounts of 1,000 ruined clones. The Setup We're using Polars for lightning-fast data processing, NumPy to simulate millions of random draws, and Matplotlib to visualize our financial doom. First, grab our historical Eurojackpot database and drop it into the same folder. The Script Here is the complete Python code. It calculates the exact mathematical odds on the fly using combinatorics (math.comb ), links them up with the real average payouts from recent years, and then runs 1,000 players through the brute-force meat grinder. """ Eurojackpot Monte Carlo Simulation =================================== Simulates realistic lottery participation over multiple years using exact combinatorial probabilities and historical payout data. """ from future import annotations import math import logging from dataclasses import dataclass, field from pathlib import Path from typing import TYPE_CHECKING import matplotlib.pyplot as plt import matplotlib.ticker as ticker import numpy as np import polars as pl if TYPE_CHECKING: pass # --------------------------------------------------------------------------- # Logging # --------------------------------------------------------------------------- logging.basicConfig(level=logging.INFO, format="%(message)s") log = logging.getLogger(name) # --------------------------------------------------------------------------- # Constants # --------------------------------------------------------------------------- MAIN_POOL = 50 MAIN_PICK = 5 EURO_POOL = 12 EURO_PICK = 2 TICKET_COST = 2.0 DRAWS_PER_YEAR = 104 # 2× per week PRIZE_TIERS: dict[str, tuple[int, int]] = { "5 + 2": (5, 2), "5 + 1": (5, 1), "5 + 0": (5, 0), "4 + 2": (4, 2), "4 + 1": (4, 1), "4 + 0": (4, 0), "3 + 2": (3, 2), "2 + 2": (2, 2), "3 + 1": (3, 1), "3 + 0": (3, 0), "1 + 2": (1, 2), "2 + 1": (2, 1), } # --------------------------------------------------------------------------- # Config dataclass # --------------------------------------------------------------------------- @dataclass class SimConfig: players: int = 1_000 years: int = 10 fields_per_draw: int = 8 output_dir: Path = field(default_factory=lambda: Path(".")) @property def total_draws(self) -> int: return self.years * DRAWS_PER_YEAR @property def cost_per_draw(self) -> float: return self.fields_per_draw * TICKET_COST @property def total_spent(self) -> float: return self.total_draws * self.cost_per_draw # --------------------------------------------------------------------------- # Probability calculation # --------------------------------------------------------------------------- def calculate_exact_probabilities() -> dict[str, float]: """Exact Eurojackpot winning probabilities via combinatorics.""" total_main = math.comb(MAIN_POOL, MAIN_PICK) total_euro = math.comb(EURO_POOL, EURO_PICK) probabilities: dict[str, float] = {} prob_sum = 0.0 for tier, (match_m, match_e) in PRIZE_TIERS.items(): ways_main = math.comb(MAIN_PICK, match_m) * math.comb( MAIN_POOL - MAIN_PICK, MAIN_PICK - match_m ) ways_euro = math.comb(EURO_PICK, match_e) * math.comb( EURO_POOL - EURO_PICK, EURO_PICK - match_e ) prob = (ways_main / total_main) * (ways_euro / total_euro) probabilities[tier] = prob prob_sum += prob probabilities["No Win"] = 1.0 - prob_sum return probabilities # --------------------------------------------------------------------------- # Data loading # --------------------------------------------------------------------------- def load_empirical_payouts( file_path: str | Path, probabilities: dict[str, float], ) -> tuple[np.ndarray, np.ndarray]: """Load historical payout data and align with probability tiers.""" log.info("[1/3] Loading historical payouts…") df = pl.read_json(str(file_path)) payouts_df = df.select(pl.col("prizeDistribution")).unnest("prizeDistribution") avg_payouts: dict[str, float] = payouts_df.mean().to_dicts()[0] tiers = list(probabilities.keys()) payouts = np.array( [0.0 if t == "No Win" else float(avg_payouts.get(t) or 0.0) for t in tiers], dtype=np.float64, ) probs = np.fromiter( (probabilities[t] for t in tiers), dtype=np.float64, count=len(tiers) ) probs /= probs.sum() # normalise to guard against floating-point drift return payouts, probs # --------------------------------------------------------------------------- # Monte Carlo simulation # --------------------------------------------------------------------------- def run_monte_carlo( payouts: np.ndarray, probs: np.ndarray, cfg: SimConfig, *, seed: int | None = None, ) -> pl.DataFrame: """ Vectorised Monte Carlo simulation. Memory layout: (total_draws, players, fields_per_draw) All draws are sampled in a single call for maximum throughput. """ rng = np.random.default_rng(seed) log.info("[2/3] Simulating %d players over %d draws…", cfg.players, cfg.total_draws) log.info( " Fields/draw: %d | Cost/draw: €%.2f | Total per player: €%.2f", cfg.fields_per_draw, cfg.cost_per_draw, cfg.total_spent, ) # --- sample outcomes for every (draw, player, field) triple at once --- outcome_indices = rng.choice( len(payouts), size=(cfg.total_draws, cfg.players, cfg.fields_per_draw), p=probs, ) raw_winnings: np.ndarray = payouts[outcome_indices] # sum over fields, subtract per-draw cost, cumsum over time net_per_draw = raw_winnings.sum(axis=2) - cfg.cost_per_draw # (draws, players) bankroll = np.cumsum(net_per_draw, axis=0) # (draws, players) return pl.DataFrame( bankroll, schema=[f"Player_{i}" for i in range(cfg.players)], ) # --------------------------------------------------------------------------- # Visualisation helpers # --------------------------------------------------------------------------- _PALETTE = { "jackpot": "#00E5FF", "major": "#FFD700", "minor": "#FF9800", "profit": "#00ff88", "loss": "#ff4d4d", "worst": "#ffffff", "breakeven": "#00bfff", "bg_fig": "#0f1117", "bg_ax": "#111827", "bg_legend": "#1f2937", "bg_text": "#1f2937", "edge_text": "#ff4d4d", } def _profit_color(balance: float) -> str: if balance >= 1_000_000: return _PALETTE["jackpot"] if balance >= 100_000: return _PALETTE["major"] if balance >= 10_000: return _PALETTE["minor"] if balance > 0: return _PALETTE["profit"] return _PALETTE["loss"] def _apply_dark_style(fig: plt.Figure, ax: plt.Axes) -> None: fig.patch.set_facecolor(_PALETTE["bg_fig"]) ax.set_facecolor(_PALETTE["bg_ax"]) def _add_stats_box( ax: plt.Axes, avg: float, winners: int, players: int, fields: int, ) -> None: ax.text( 0.02, 0.04, f"Stake per draw: €{fields * TICKET_COST:.2f}\n" f"Average result: €{avg:,.0f}\n" f"Profitable players: {winners}/{players}", transform=ax.transAxes, fontsize=12, bbox=dict( boxstyle="round,pad=0.6", facecolor=_PALETTE["bg_text"], edgecolor=_PALETTE["edge_text"], alpha=0.9, ), ) def _euro_formatter(x: float, _pos: int) -> str: return f"€ {int(x):,}" # --------------------------------------------------------------------------- # Plot generation # --------------------------------------------------------------------------- def analyze_and_plot(df: pl.DataFrame, cfg: SimConfig) -> None: """Generate log-scale (illusion) and linear-scale (reality) charts.""" log.info("[3/3] Creating visualisations…") players = len(df.columns) final_balances = np.array(df.row(-1)) avg_balance = float(final_balances.mean()) best_balance = float(final_balances.max()) worst_balance = float(final_balances.min()) best_idx = int(final_balances.argmax()) worst_idx = int(final_balances.argmin()) best_col = df.columns[best_idx] worst_col = df.columns[worst_idx] winners_mask = final_balances > 0 winners_count = int(winners_mask.sum()) winner_cols = [df.columns[i] for i, w in enumerate(winners_mask) if w and df.columns[i] != best_col] loser_cols = [df.columns[i] for i, w in enumerate(winners_mask) if not w] # cap losers shown to 500 for readability sample_losers = loser_cols[:500] plt.style.use("dark_background") chart_configs: list[tuple[str, str, bool, float | None]] = [ ( "eurojackpot_1_illusion_log.png", f"The Illusion: {cfg.years} Years Eurojackpot ({cfg.fields_per_draw} Fields)", True, None, ), ( "eurojackpot_2_reality_linear.png", f"The Reality: {cfg.years} Years Eurojackpot ({cfg.fields_per_draw} Fields)", False, 5_000.0, ), ] for filename, title, use_log, ylim_max in chart_configs: fig, ax = plt.subplots(figsize=(12, 7)) _apply_dark_style(fig, ax) # losers (faint red) for col in sample_losers: ax.plot(df[col].to_numpy(), color=_PALETTE["loss"], alpha=0.15, l