The Mathematics of Winning Monopoly -

The Mathematics of Winning Monopoly

Stand-up Maths
Views: 2879363
Like: 58605
I talk to Hannah Fry and compare our mathematical investigations into playing Monopoly. I’ve put all my probabilities below.

You can buy a signed copy of Hannah’s book on Maths Gear:

In the UK you can also get it from Waterstones:

There is a Kindle edition on Amazon:

You can download all of my monopoly code here:

Here are Hannah’s plots:

– I said the game has “finite money” but Chuck Monster has correctly pointed out that the official Monopoly rules state “It [the bank] never runs out of money; if it runs out of bills, players can use any convenient items as substitutes until the bank gets enough money.”

Support my videos on Patreon:

Filming by Trunkman Productions
Music by Howard Carter
Design by Simon Wright

MATT PARKER: Stand-up Mathematician
Maths book:
Nerdy maths toys:


From 100,000,000 rolls.

00 2.854% – GO
01 2.109% – Old Kent Road/Mediterranean Avenue
03 2.261% – Whitechapel Road/Baltic Avenue
04 2.404% – Income Tax
05 2.757% – Kings Cross Station/Reading RR
06 2.317% – The Angel Islington/Oriental Avenue
07 1.017% – CHANCE
08 2.308% – Euston Road/Vermont Avenue
09 2.278% – Pentonville Road/Connecticut Avenue
10 6.325% – JAIL
11 2.719% – Pall Mall/St. Charles Place
12 2.61% – Electric Company
13 2.395% – Whitehall/States Avenue
14 2.477% – Northumberland Avenue/Virginia Avenue
15 2.804% – Marylebone Station/Pennsylvania RR
16 2.8% – Bow Street/St. James Place
18 2.94% – Marlborough Street/Tennessee Avenue
19 3.096% – Vine Street/New York Avenue
20 2.868% – Free Parking
21 2.839% – The Strand/Kentucky Avenue
22 1.212% – CHANCE
23 2.731% – Fleet Street/Indiana Avenue
24 3.197% – Trafalgar Square/Illinois Avenue
25 2.897% – Fenchurch St Station/B&O RR
26 2.719% – Leicester Square/Atlantic Avenue
27 2.689% – Coventry Street/Ventnor Avenue
28 2.821% – Water Works
29 2.601% – Piccadilly/Marvin Gardens
30 0% – GO TO JAIL
31 2.675% – Regent Street/Pacific Avenue
32 2.616% – Oxford Street/North Carolina Avenue
34 2.494% – Bond Street/Pennsylvania Avenue
35 2.553% – Liverpool Street Station/Short Line
36 2.239% – CHANCE
37 2.085% – Park Lane/Park Place
38 2.086% – Luxury Tax
39 2.552% – Mayfair/Boardwalk


  1. My way of playing is just go around the board and buy out EVERY single tile I land on.

  2. No house rules? What madness is that? It's not just the ordinary stuff like free parking payouts. Matt has it right, it's much more fun when you start doing creative financial stuff, loan-sharking each other for advantages. One of my favourite little tweaks is trading to give someone a monopoly, but placing a rent-free condition so that you can land on that set without cost.

  3. If you are playing a 4-people game, and you are the first to go. The best thing that can happen to you is hit 12. You land on utility. Even though it's crap, buy it. But you get chance to throw again. If you hit 12 again, you are now on the most visited place in monopoly. So of course you buy it. And then you hit 11. You will then end up 5 spots from getting more money, and you got one of the trains. And since you were the first to go, you are just 5 spots behind the others on go. And for the next round, buy everything you can get the chance. But have a main goal of getting the reds. If some other buy the other utility, yours is worthless, so just flip it down to get some cash.

  4. It's actually a myth that comments in your code are good, they are not. In bigger programs, code tends to get updated, while comments tend to stay; and after a while comment says one thing, but code does the other. The way better method of writing good code, is to make it more readable, so that the comments aren't needed, and that's hard.

  5. PS: money in monopoly actually never ends.

  6. Love to see a book on the original monopoly before it was monopolized

  7. Interesting that it isnt always the 2nd color set on a side that is the best target. Youd think it wouldnt be that way because you are paying the same amount for houses but get more money back.

  8. Always buy the Orange set if you can; and then add the Purple one as well if you can. Guaranteed to win. I learnt this when I was 10…….

  9. Obviously, the optimal way to play monopoly is with Hannah Fry.

  10. Orange and red are the king properties! I aim for those and will make questionable trades just to get them. They pay back generously once developed. Otherwise, I like light blue because you can take people’s “Go” money. 😉

  11. You should try Monopoly stock exchange. Each road is a company and you can buy 7 shares in each one. You are only allow to buy 4 shares each time. A calculator takes into account the money that each shareholder has to pay each time and the shares are changing in price.

  12. It’s sometimes Park Place and sometimes Park Lane? Sounds like a real Parker Square

  13. I have a few things my friends and I do to have short games: All taxes go into a pool to be collected when someone lands on free parking. Play aggressively, trade when it make sense to, no auctions, and place houses or hotels whenever you can. Also the first side of the board is really strong if you own it all. Especially if you own the first half of the board.

  14. 5:46 true statement👍🏼

    You should make these kind of analysis with modern board games. It would be great.

  15. A proper game will never end, because the net money going towards the players is positive due to the "start earnings". Hence, with a good distribution of streets between i.e. 2 players both will make a fortune.

  16. "Matt, I suggest a new strategy: let the redhead win…"

  17. with more than two players you need to change strategy. Houses become scarce so its good to hoard them, the player with the $50 houses and another set will win because he has the ability to develop to 4 houses (not hotels keep the houses and stop other players expanding its dirty but its play to win)

  18. Just playing a lot as a youngster the Oranges are the streets to go for and the 4 stations. Also do not underestimate Old Kent Road and Whitechapel.

  19. I'm quite successful winning with the yellow city (Den Haag). The house prices are the same for each side the yellow should be better than red. Also the lower cities are not winning so fast, so that you can arrive at collecting yellow. As people follow Hannah's rule there comes mixed game strategies into play. With meta, meta-meta etc. computers cannot beat human players in poker. A monopoly playing contest with programs shouldbe interesting.

  20. Why does the relative attractiveness of a person in relation to their intelligence differ drastically between genders?

    Marry me Hannah!

  21. Thank you! Fascinating. Setting up mics for both speakers would be helpful. I's not pleasant, hiring and lowering the volume as the conversation changes tones.

  22. Can't you use an eigendecomposition to solve this?

  23. Math always oversimplifies things……(barks the 17th century Russian literature grad)!
    What about the brutal opportunity cost' of buying a high end property early in the game that cripples the players cash flow and handcuffs them from doing anything else….like buying property with a near term return and NOT LOSING. Doesn't the stage of the game influence some decisions with variables that are beyond simple player landing zone distribution???? What about the significance of the family's 'birth order' power dynamic and that influence on player trades? What about the variable of an aunt with a heroic drinking problem. Id like the see the python coding for that! Admittedly……i have no supporting data to counter ….but I the stand by my rhetoric and you should trust me.

  24. I feel skeptical of the outcome. There are additional factors in monopoly to take into account. For example, the probability of getting a set changes a lot once you've already landed on a member of that set–and that's when you need to decide whether to invest in it, not before the game starts.

    Also, green may have really good returns, but even if you're lucky enough to land on all three (instead of overpaying through trades for them), hotels on all three are the most expensive in the game at over $4k (including property). Of course you don't need hotels, but the first meaningful payment is at 2 houses which is still > $2300. And 3 houses, the optimum investment, is nearly $3k. So using the green strategy basically requires that nobody else get a decent set beforehand.

    Also also, ML algorithms for this game actually favor railroads if you can get 3 or 4 of them–4 positions of $200 rent with multiple probability-boosting CC/chance cards is pretty great.

    So I'm skeptical that this advice is comprehensive.

  25. I agree with Hannah, it is always nice to have an excuse to do some matrix multiplication.

  26. Last time I played Monopoly I applied some maths and probabilities to my strategy, I did win (although we stopped before I officially won, but I was leading!) but one major spanner was thrown into my plan. I had my eyes set on the orange and red properties because I checked the probability of landing on each set in advance and these were the most likely, however, other players, none the wiser to my strategy, just so happened to beat me to some of those properties and incidentally, bought them! This messed me up a bit, although I took my chances and WAY overpaid them and bought them back off these other players, effectively putting all my eggs in the maths basket and hoping it would pay off, fortunately it did!
    But it would have been cool for them to discuss how to optimise your strategy when this happens. It’s all well and good saying “maths say buy these sets ASAP!” But if you’re playing with 4-5 others, odds are they’re going to scoop up the properties you want before you get to them! Although I suppose they didn’t talk about it because it’s not really possible to calculate this, as it’s unpredictable human behaviour… still though, really enjoyed the video! 🙂

  27. I am fascinated that when trying to simulate 2d6, instead of just rolling 2d6 separately and checking for doubles, Matt's first thought was to create a d36 where each side represents one of the potential 2d6 rolls.

  28. Oh capitalism is soo bad…

    Central planning economy is what's good, "innit comrade"??

  29. 14:43 that was funny when Hannah copied Matt hitting the table

  30. my strategy to win monopoly, put 3 houses on each of your streets, that's it.

  31. My friends always wonder why I take a 19-minute long bathroom break before playing Monopoly.

  32. no game is more violet then this game with your family … the RAGE …last the whole year…

  33. The entire video is just advertisement for the shitty little book.

  34. The call out about the American dream collapsing while the media gaslights us earned the like. Capitalism is trash.

  35. Mathematician is just a bunch of people with hyper curiosity and playing around just like a little kid, definitely!

  36. can you tell how the infinite happens

  37. If you don't get lucky dice you will never win this game;

  38. Not a python developer but is that code really that bad?

  39. Good now it will only take a few monthes 🙂

Leave a Reply

Your email address will not be published.