The deck and the odds

Appendix B online. An interactive odds calculator is coming; the tables are exact today.

The card list, and the arithmetic you can do from it. Every number here is exact combinatorics on the 106-card deck, except the last table, which is measured. The draw odds ignore Pass Go, the five-card draw on an empty hand, and reshuffles — all of which put more cards through more hands sooner — so they are lower bounds on what you will see in a real game.

The deck

106 playing cards, plus four rules cards. Face value in the whole deck: $252. Mean card: $2.38. A five-card opening hand is worth about $11.89 if banked outright; a two-card draw, $4.75.

Class Cards Value each Total
Money 20 6 × $1, 5 × $2, 3 × $3, 3 × $4, 2 × $5, 1 × $10 $57
Property 28 by colour, below $67
Property wild 11 Light Blue/Brown $1; Light Blue/Railroad $4; Pink/Orange $2 (×2); Red/Yellow $3 (×2); Dark Blue/Green $4; Green/Railroad $4; Railroad/Utility $2; rainbow $0 (×2) $25
Rent 13 five two-colour pairs, 2 each at $1; 3 any-colour at $3 $19
Action 34 2 Deal Breaker $5; 3 Just Say No $4; 3 Sly Deal $3; 3 Forced Deal $3; 3 Debt Collector $3; 3 It's My Birthday $2; 2 Double the Rent $1; 3 House $3; 2 Hotel $4; 10 Pass Go $1 $84

Only 20 cards are money and nothing else, while another 47 — the actions and the rents — are money that also does something; the two rainbow wilds are worth nothing at all and cannot pay a debt.

Colour Set size Cards Value each Rent by cards held
Brown 2 2 $1 1, 2
Light Blue 3 3 $1 1, 2, 3
Pink 3 3 $2 1, 2, 4
Orange 3 3 $2 1, 3, 5
Red 3 3 $3 2, 3, 6
Yellow 3 3 $3 2, 4, 6
Green 3 3 $4 2, 4, 7
Dark Blue 2 2 $4 3, 8
Railroad 4 4 $2 1, 2, 3, 4
Utility 2 2 $2 1, 2

A House adds $3 to a full set's rent and a Hotel $4 more; neither goes on Railroads or Utilities.

Your first cards

Chance of holding at least one of the card, by how many cards you have seen. Five is the deal, and seven, nine and eleven are the start of your first, second and third turns if you go first and draw no extras.

At least one… 5 cards 7 9 11
Just Say No (3 in deck) 13.6% 18.7% 23.6% 28.3%
Deal Breaker (2) 9.3% 12.8% 16.3% 19.8%
Sly or Forced Deal (6) 25.7% 34.3% 42.1% 49.1%
Debt Collector or Birthday (3 each) 13.6% 18.7% 23.6% 28.3%
Any Rent card (13) 48.7% 61.1% 70.7% 78.1%
Any-colour rent (3) 13.6% 18.7% 23.6% 28.3%
Pass Go (10) 39.7% 51.1% 60.5% 68.3%
Money card of $4 or more (6) 25.7% 34.3% 42.1% 49.1%
The $10 (1) 4.7% 6.6% 8.5% 10.4%

How to use this: read down to the card you are hoping for and across to how many cards you have seen. The number is the chance you hold at least one.

A complete set in the deal is rare: 1.7% for any given two-card colour, 0.17% for a three-card colour, 0.01% for the Railroads, wilds counted.

Somebody at the table has it

Chance that at least one copy was dealt to someone, by player count.

At least one… 2 players 3 4 5
Just Say No 25.9% 37.0% 47.0% 55.8%
Deal Breaker 18.1% 26.4% 34.3% 41.8%
Sly or Forced Deal 45.7% 60.9% 72.4% 81.0%
Any Rent card 74.6% 88.0% 94.5% 97.6%
The $10 9.4% 14.2% 18.9% 23.6%

How to use this: before the first turn, this is the chance that somebody at your table was dealt the card. At four players a Just Say No is out there about half the time.

When the big cards arrive

Expected round (every player has taken that many turns; the deal is round 0) by which a card has entered someone's hand.

Card 2 players 3 4 5
First Deal Breaker 6.9 4.1 2.7 1.9
Second Deal Breaker 15.7 9.8 6.9 5.1
First Just Say No 4.8 2.7 1.7 1.1
Third Just Say No 17.9 11.3 8.0 6.0
The $10 11.3 6.9 4.8 3.5

At four players the first Deal Breaker is in a hand by the end of round 3 two times in three, and by the end of round 5 four times in five. Without Pass Go the deck runs out at the end of round 24, 16, 11, or 9 for two to five players. With ten Pass Go cards and empty-hand draws it runs out sooner, which is why four- and five-player games so often reshuffle.

Set economics

"Servers" counts the natural cards plus every wild that can show the colour (the two rainbows included). "Tied up" is the face value you give up by laying the full set instead of paying with it.

Colour Size Servers $ tied up Full rent Rent per $ With House + Hotel Exposure while building
Brown 2 5 $2 $2 1.00 $9 one card, one Sly Deal
Light Blue 3 7 $3 $3 1.00 $10 two cards
Pink 3 7 $6 $4 0.67 $11 two cards
Orange 3 7 $6 $5 0.83 $12 two cards
Red 3 7 $9 $6 0.67 $13 two cards
Yellow 3 7 $9 $6 0.67 $13 two cards
Green 3 7 $12 $7 0.58 $14 two cards
Dark Blue 2 5 $8 $8 1.00 $15 one card, one Sly Deal
Railroad 4 9 $8 $4 0.50 no buildings three cards
Utility 2 5 $4 $2 0.50 no buildings one card, one Sly Deal

Dark Blue is the outlier: the highest rent per card in the deck, from two cards, with only five servers, while Green costs the most to lock up, $12 of property for a $7 rent. The two-card colours are complete after a single card of exposure, and the Railroads, which need four of their nine servers, can never carry a building at all. Which colour to lay first is priced in Chapter 6, and the answer there is the set you are closest to finishing, whatever it rents.

What each card is worth

This last table is not arithmetic; it is measured. Forbid one card class to a single seat, leave everything else about the table alone, and the drop in that seat's win rate is what the class was worth to it. The first four columns are a folk seat at a folk table (E03); the last four are a strong seat at a strong table, which is the harder and more useful question (E45). All figures are points of win rate, at 40,000 and 100,000 games a cell.

Card class Folk 2p 3p 4p 5p Strong 2p 3p 4p 5p
Rent cards 13.1 16.7 18.3 16.5 7.0 12.7 14.6 14.3
Pass Go 19.4 16.6 12.5 10.0 12.4 11.6 10.3 9.2
Deal Breaker 14.3 12.2 8.5 6.3 20.6 14.0 10.9 8.6
Sly Deal 8.1 7.8 5.2 3.2 9.1 7.3 5.5 4.2
Just Say No 7.8 5.6 3.2 2.2 11.6 8.1 5.3 3.6
Debt Collector 8.6 5.8 3.8 2.4 5.8 5.7 4.6 3.5
It's My Birthday 4.8 5.6 5.5 5.1 2.4 4.4 5.9 6.5
Forced Deal 2.1 2.9 1.7 0.8 4.6 4.2 3.1 2.3
Moving wild cards 1.4 1.9 0.7 0.7 1.8 1.8 0.9 0.5
Double the Rent 1.0 0.9 0.0 −0.2 0.9 1.0 0.9 0.6
House and Hotel 0.9 1.0 0.3 0.0 −0.4 −0.6 −0.6 −0.6

Where two classes sit within a tenth or two of each other the table is not ranking them, because the measurement cannot: at three players the Birthday and the Forced Deal, at four the Sly Deal and the Just Say No and the wilds and Double the Rent, at five the Just Say No and the Debt Collector and again the wilds and Double the Rent. Every value here comes from 100,000 games a cell.

How to use this: read the right-hand half. It says which cards reward skill. Where the strong column beats the folk column, the card is worth learning; where it does not, the card plays itself.

The two halves tell different stories, and the gap between them is the book in miniature. A strong player gets more out of every card that has to be aimed and timed: the Deal Breaker, which becomes the most valuable card in the deck head to head at 21.1 points; the Just Say No; the Sly Deal; and above all the Forced Deal, worth more than twice as much in good hands as in ordinary ones. It gets less out of the blunt money cards, because it was not wasting the good ones to begin with — rent falls from 13.1 to 7.2 head to head, and the Debt Collector and the head-to-head Birthday fall with it. Buildings are the extreme case. To a folk seat they are worth about a point; to a strong seat they are worth nothing, and at four and five players slightly less than nothing, which means the strong player does better banking a House than placing it. Under the 2024 sheet, a building on a set that can be broken is not an asset. It is a hostage.