--- title: Monopoly Miscalculations: Mathematical Modeling with... description: A deep-dive compares two approaches to modeling Markov chains: Monte Carlo simulation and transition matrix (eigenvector) analysis, first applied to a simple... canonical: https://daily.dev/posts/monopoly-miscalculations-mathematical-modeling-with-markov-chains-monte-carlo-simulations-and-mat-myvkmv4xj twitter:card: summary_large_image twitter:site: @dailydotdev og:type: website og:site_name: daily.dev og:title: Monopoly Miscalculations: Mathematical Modeling with Markov Chains, Monte Carlo Simulations, and Matrix Methods | daily.dev og:description: A deep-dive compares two approaches to modeling Markov chains: Monte Carlo simulation and transition matrix (eigenvector) analysis, first applied to a simple... og:url: https://daily.dev/posts/monopoly-miscalculations-mathematical-modeling-with-markov-chains-monte-carlo-simulations-and-mat-myvkmv4xj og:image: https://api.daily.dev/og/posts/mYVKmv4xJ.png og:image:alt: Monopoly Miscalculations: Mathematical Modeling with Markov Chains, Monte Carlo Simulations, and Matrix Methods og:image:width: 1200 og:image:height: 630 og:locale: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # Monopoly Miscalculations: Mathematical Modeling with Markov Chains, Monte Carlo Simulations, and Matrix Methods **[EmbeddedRelated](https://daily.dev/sources/embeddedrelated)** · 52 min read · 0 upvotes · 0 comments ## Summary A deep-dive compares two approaches to modeling Markov chains: Monte Carlo simulation and transition matrix (eigenvector) analysis, first applied to a simple bank queueing example, then to the board game Monopoly. Using NumPy and SciPy, it builds an 8-state transition matrix for the queue and a 120-state transition matrix for Monopoly (accounting for dice rolls, doubles, Jail, and Chance/Community Chest cards), then computes stationary distributions via power iteration, eigenvalue decomposition, and null-space methods to determine the steady-state probability of landing on each board space. Code examples and tables walk through state definitions, transition probabilities, and convergence rates. ## Full article daily.dev links to this article rather than hosting it. Read it at the original source: ## Questions this post answers ### How many states are needed to model Monopoly as a Markov chain accounting for dice rolls and jail? 120 states are needed rather than the 40 board spaces, because the model must also track the history of doubles rolled during a turn. There are 39 possible landing spaces (excluding the always-vacated Go to Jail space) times 3 doubles-count states, plus 3 additional states for the three possible turns spent in Jail, giving 39 × 3 + 3 = 120 total states. _Developers building probabilistic simulations can find similar state-space modeling walkthroughs curated on daily.dev._ ### What is the difference between Monte Carlo simulation and transition matrix analysis for Markov chains? Monte Carlo simulation estimates behavior by running many pseudorandom trials and averaging results, with accuracy improving only as the square root of the number of samples, requiring about a million trials for tenfold better precision. Transition matrix analysis instead computes the exact stationary distribution directly as the eigenvector of the transition matrix corresponding to eigenvalue 1, avoiding sampling uncertainty entirely for ergodic chains. _Engineers deciding between simulation and analytical methods can compare trade-offs like these through daily.dev._ ### Why is Jail the most commonly landed-on space in Monopoly? Jail accumulates probability because players are sent there directly by the Go to Jail space, by Chance and Community Chest cards, and by rolling doubles three times in a row, while players already in Jail have only a small chance of leaving via double-six. This funnel effect, combined with the oranges and reds being common landing spots right after leaving Jail, makes Jail and the nearby orange/red properties disproportionately likely. _Players and modelers analyzing game probability dynamics can track similar statistical breakdowns on daily.dev._ ## Similar posts on daily.dev - [Markov Chains: nuclear algorithm used by Quants to crush 87% of Polymarket traders](https://daily.dev/posts/markov-chains-nuclear-algorithm-used-by-quants-to-crush-87-of-polymarket-traders-czdce9mxh) · Coins Bench · 2 upvotes · 1 comments - [Monte-Carlo simulations](https://daily.dev/posts/monte-carlo-simulations-svqxefotg) · Eli Bendersky · 0 upvotes · 0 comments - [The Entropy of a Markov Chain](https://daily.dev/posts/the-entropy-of-a-markov-chain-mehuhcdab) · Hacker News · 0 upvotes · 0 comments - [Markov Decision Processes and Value Functions in RL](https://daily.dev/posts/markov-decision-processes-and-value-functions-in-rl-50qdbum2g) · Daily Dose of Data Science \| Avi Chawla \| Substack · 2 upvotes · 0 comments - [Bellman Equation: A Comprehensive Guide](https://daily.dev/posts/bellman-equation-a-comprehensive-guide-nfsuexrh9) · Medium · 0 upvotes · 0 comments --- Tags: [#python](https://daily.dev/tags/python), [#data-science](https://daily.dev/tags/data-science), [#math](https://daily.dev/tags/math), [#numpy](https://daily.dev/tags/numpy) [View this post on daily.dev](https://daily.dev/posts/monopoly-miscalculations-mathematical-modeling-with-markov-chains-monte-carlo-simulations-and-mat-myvkmv4xj) ```json {"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://daily.dev/#organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180},"sameAs":["https://twitter.com/dailydotdev","https://github.com/dailydotdev","https://www.linkedin.com/company/daily-dev-ltd"]},{"@type":"WebSite","@id":"https://daily.dev/#website","url":"https://daily.dev","name":"daily.dev","publisher":{"@id":"https://daily.dev/#organization"},"potentialAction":{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://daily.dev/search?q={search_term_string}"},"query-input":"required name=search_term_string"}}]} {"@context":"https://schema.org","@type":"TechArticle","headline":"Monopoly Miscalculations: Mathematical Modeling with Markov Chains, Monte Carlo Simulations, and Matrix Methods","url":"https://daily.dev/posts/monopoly-miscalculations-mathematical-modeling-with-markov-chains-monte-carlo-simulations-and-mat-myvkmv4xj","mainEntityOfPage":{"@type":"WebPage","@id":"https://daily.dev/posts/monopoly-miscalculations-mathematical-modeling-with-markov-chains-monte-carlo-simulations-and-mat-myvkmv4xj"},"datePublished":"2026-10-03T10:09:17.044Z","dateModified":"2026-10-03T10:09:42.874Z","description":"A deep-dive compares two approaches to modeling Markov chains: Monte Carlo simulation and transition matrix (eigenvector) analysis, first applied to a simple...","image":"https://media.daily.dev/image/upload/s--ZrL_HSsR--/f_auto/v1722860399/public/Placeholder%2006","thumbnailUrl":"https://media.daily.dev/image/upload/s--ZrL_HSsR--/f_auto/v1722860399/public/Placeholder%2006","isAccessibleForFree":true,"articleSection":"EmbeddedRelated","inLanguage":"en","publisher":{"@type":"Organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180}},"author":{"@type":"Organization","name":"EmbeddedRelated","logo":"https://media.daily.dev/image/upload/s--OVGOU0Ec--/f_auto,q_auto/v1780213554/logos/embeddedrelated?_a=BAMAMiWQ0","url":"https://daily.dev/sources/embeddedrelated"},"commentCount":0,"discussionUrl":"https://daily.dev/posts/monopoly-miscalculations-mathematical-modeling-with-markov-chains-monte-carlo-simulations-and-mat-myvkmv4xj","interactionStatistic":[{"@type":"InteractionCounter","interactionType":{"@type":"LikeAction"},"userInteractionCount":0},{"@type":"InteractionCounter","interactionType":{"@type":"CommentAction"},"userInteractionCount":0}],"keywords":"python,data-science,math,numpy","timeRequired":"PT52M"} {"@context":"https://schema.org","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://daily.dev"},{"@type":"ListItem","position":2,"name":"EmbeddedRelated","item":"https://daily.dev/sources/embeddedrelated"},{"@type":"ListItem","position":3,"name":"Monopoly Miscalculations: Mathematical Modeling with Markov Chains, Monte Carlo Simulations, and Matrix Methods"}]} {"@context":"https://schema.org","@type":"FAQPage","@id":"https://daily.dev/posts/monopoly-miscalculations-mathematical-modeling-with-markov-chains-monte-carlo-simulations-and-mat-myvkmv4xj#faq","mainEntity":[{"@type":"Question","name":"How many states are needed to model Monopoly as a Markov chain accounting for dice rolls and jail?","acceptedAnswer":{"@type":"Answer","text":"120 states are needed rather than the 40 board spaces, because the model must also track the history of doubles rolled during a turn. 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