Probability Guide

Heart of Crown looks like a game of chance, but it is fundamentally a game of controlling probability.

"I kept drawing estates all game." "My Stargazing Witch never showed up." Every player has felt this frustration. Yet experienced players manage this "randomness" through mathematics.

The essence of a deck-building game is manipulating probability distributions in your favor. Trashing estates is a mathematical operation that "reduces the probability of drawing weak cards." Adding draw cards is a statistical operation that "increases the sample size per turn."

This article explains the probabilistic aspects of Heart of Crown using formulas and concrete numbers. The goal is intuitive understanding, not memorizing formulas.

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The Hypergeometric Distribution

The probability of drawing a specific card in a card game follows the hypergeometric distribution — the same problem as "given a box of N balls where k are red, what is the probability of drawing at least one red ball when you draw h balls?"

Formula

P = 1 - C(N-k, h) / C(N, h)

• N = total cards in deck
• k = number of target cards
• h = draw size (usually 5)
• C(a, b) = combinations (ways to choose b from a)

The "1 - X" structure means "1 minus the probability of drawing zero copies" = probability of drawing at least one.

Concrete Examples (10-card deck, 5-card draw)

1 copy of target card: P = 1 - C(9,5)/C(10,5) = 1 - 126/252 = 50%

With only 1 copy in a 10-card deck, the card appears in half of all turns.

2 copies of target card: P = 1 - C(8,5)/C(10,5) = 1 - 56/252 = 77.8%

Two copies gives nearly an 80% chance each turn.

3 copies of target card: P = 1 - C(7,5)/C(10,5) = 1 - 21/252 = 91.7%

Three copies ensures the card appears in over 90% of turns. This is the mathematical basis for "buy 3 copies of the same card."

Intuitive Approximation

Without complex formulas, use this rule:

"Draw probability ≈ (copies ÷ deck size) × 5"

Example: 12-card deck, 3 Cities → (3 ÷ 12) × 5 = 1.25 → roughly 80% chance of drawing at least 1

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Comprehensive Probability Table

The following shows "probability of drawing at least 1 copy of a card with n copies, in a 5-card draw."

Deck Size	1 copy	2 copies	3 copies	4 copies
6 cards	83.3%	100%	100%	100%
8 cards	62.5%	89.3%	98.2%	100%
10 cards	50.0%	77.8%	91.7%	97.6%
12 cards	41.7%	65.9%	82.6%	92.4%
15 cards	33.3%	55.6%	72.5%	84.6%
20 cards	25.0%	43.4%	58.4%	70.8%

Key insight: Doubling the deck size roughly halves the probability of drawing a specific card. Keeping the deck thin is the foundation of probabilistic stability.

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How Each Estate Removed Improves Your Deck

Starting deck: 7 Estates (1 coin each) + 3 Handmaidens (0 coins) = 10 cards total

Estates remaining	Deck size	Estate density	Expected estates in 5-card draw
7 (starting)	10	70%	3.50
6	9	66.7%	3.33
5	8	62.5%	3.13
4	7	57.1%	2.86
3	6	50.0%	2.50
2	5	40.0%	2.00
1	4	25.0%	1.25
0	3	0%	0

Why Estate Removal Stabilizes Your Deck (Three Mathematical Reasons)

Reason 1: Relative increase in useful card draw probability

With a 10-card deck containing 3 useful cards: expected draw = 3/10 × 5 = 1.5 per turn After removing 2 estates (8-card deck, same 3 useful cards): 3/8 × 5 = 1.875 per turn

That is a 25% improvement from trashing just 2 estates.

Reason 2: Increased deck cycling speed

Thinner decks complete a full cycle in fewer turns (see next section), meaning newly purchased cards become available faster.

Reason 3: Avoiding -2 succession point penalty

Estates carry -2 succession points. Leaving many estates in your deck when the scoring phase arrives significantly hurts your final total.

Estate Removal Priority Table

Situation	Removal cost	Economic improvement	Priority
Turns 1-3 (7 estates)	Donation: 2 cost	Very high	S
Turns 4-6 (4-6 estates)	Donation: 2 cost	High	A
Turn 7+ (1-3 estates)	Donation: 2 cost	Moderate	B
0 estates remaining	—	Not needed	—

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Basic Cycle Calculation

Cycling speed (turns per cycle) = Deck size ÷ 5

Deck size	Turns per cycle	Cycles in 10 turns
6 cards	1.2 turns	8.33×
8 cards	1.6 turns	6.25×
10 cards	2.0 turns	5.0×
12 cards	2.4 turns	4.17×
15 cards	3.0 turns	3.33×
20 cards	4.0 turns	2.5×

Benefits of Faster Cycling

1. Cards you buy become available sooner

• 20-card deck: wait ~2 turns before using a newly bought card
• 10-card deck: wait ~1 turn
• 6-card deck: almost always available next turn

2. More attempts at key combinations

Faster cycling means more opportunities per game to assemble key combinations like City + Stargazing Witch + Alchemist in the same hand.

3. Earlier coronation

The more quickly your deck cycles, the sooner you reach a hand that satisfies coronation requirements.

Effect of Draw Cards on Cycling Speed

Draw card	Extra draws	Improvement in 10-card deck
Alchemist (cost 5, +2 draw) × 1	+2 per play	~9% faster cycling
Alchemist × 2	+4 maximum	~18% faster
Fast Horse (cost 2, +1 draw) × 1	+1 per play	~4% faster
Supply Corps (cost 4, +2-3 draw) × 1	+2-3 per play	~12% faster

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Starting Deck Expected Coins

Starting deck: 7 Estates (1 coin) + 3 Handmaidens (0 coin) = 7 total coins

Expected coins in 5-card draw: = (7 ÷ 10) × 5 = 3.5 coins

On average, the starting deck produces only 3.5 coins per turn. A City costs 4 coins, so you'll be short roughly every other turn early on.

Expected Coins as You Add Cities

Cities added	Deck size	Total coins	Expected coins/turn
0 (starting)	10	7	3.50
+1	11	9	4.09
+2	12	11	4.58
+3	13	13	5.00
+4	14	15	5.36
+5	15	17	5.67

Combined Effect: Cities + Estate Removal

State	Deck size	Total coins	Expected coins/turn
Starting	10	7	3.50
+1 City, no removal	11	9	4.09
+1 City, -1 Estate	10	9	4.50
+2 Cities, -2 Estates	10	11	5.50
+3 Cities, -3 Estates	10	13	6.50
+3 Cities, -5 Estates	8	13	8.13
+1 Grand City, +3 Cities, -7 Estates	7	15	10.71

Combining estate removal with city purchases dramatically improves expected coin output compared to simply stacking cities.

Required Coin Density for Reliable 12-Coin Turns

Coronation requires 12 coins. For the expected value to reach 12 with a 5-card draw:

(Total coins ÷ Deck size) × 5 ≥ 12 → Coin density ≥ 2.4

Reaching coin density 2.4 requires a very efficient deck — typically: 0 estates, multiple Grand Cities, and several Cities. This is why the late-game strategy centers on accumulating high-value economic cards while maintaining a thin deck.

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Cumulative Probability Calculation

If the probability of meeting coronation conditions in a single turn is p, the probability of achieving coronation within X turns is:

1 - (1 - p)^X

p = 25% per turn:

Turn	Cumulative probability
4	68.4%
6	82.2%
8	90.0%
10	94.4%
12	96.8%

p = 20% per turn:

Turn	Cumulative probability
6	73.8%
8	83.2%
10	89.3%
12	93.1%

Key insight: Even if each individual turn's coronation probability is low, repeatedly attempting it raises cumulative probability significantly. This is why completing your deck quickly and attempting coronation every turn is the right strategy.

Stargazing Witch's Effect on Coronation Probability

The Stargazing Witch (cost 3) lets you see and rearrange your deck top. This allows deterministic placement of your Grand City for the following turn, effectively boosting per-turn coronation probability by 30-50% for that specific turn.

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Cost Efficiency Comparison

Card	Cost	Points	Efficiency (pts/cost)
Handmaiden	3	2	0.667
Senator	5	3	0.600
Margrave	6	3 (+special)	0.500-0.667
Duke	8	6	0.750
Emperor's Crown	13	14	1.077

Emperor's Crown vs Duke: Decision Analysis

Option	Total cost	Total points	Efficiency	Feasibility
Emperor's Crown × 1	13	14	1.077	Requires 13-coin turn
Duke × 2	16	12	0.750	Moderate difficulty
Duke + Senator × 2	18	12	0.667	Easier

Emperor's Crown has superior raw efficiency, but the feasibility of a 13-coin turn must be factored in. In most games, buying multiple Dukes is more practical than attempting Emperor's Crown.

Optimal Purchases by Remaining Turns

Remaining turns	Recommended priority	Reason
10+	Duke > Emperor's Crown	Time permits high-cost attempts
6-9	Duke > Senator	Balance consistent point accumulation
3-5	Senator > Margrave	Prioritize affordable cards
1-2	Buy whatever you can	Maximize points with available coins

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Example 1: Comeback Calculation

Situation: You have 15 succession points. Opponent has 22. Estimated 3 turns remain.

Your deck (9 cards, expected 7 coins/turn) can reliably afford Dukes (8 coins, 6 points).

• Turn 1: Buy Duke → +6 pts → total 21
• Turn 2: Buy Duke → +6 pts → total 27
• Turn 3: Buy Senator → +3 pts → total 30

If opponent also collects 4 points/turn: 22 + 12 = 34 points.

Conclusion: Need 4+ more points. Use Stargazing Witch to stabilize Duke purchases, or leverage Margrave's special effect for extra points.

Example 2: Donation vs City Purchase

Situation: Turn 3. Hand: 3 coins (3 Estates drawn). Supply has Donation (cost 2) and City (cost 4).

With 3 coins, you cannot afford City. Buying Donation to trash an Estate is the correct play — it improves your deck's long-term efficiency at minimal cost.

Rule of thumb: 3 or fewer coins → prioritize Donation; 4-5 coins → consider City; 6+ coins → aim for Grand City.

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Key Numbers to Remember

Situation	Probability / Value
10-card deck, 1-copy card draw chance	50%
10-card deck, 3-copy card draw chance	91.7%
Starting deck expected coins/turn	3.5
Required coin density for stable 12-coin turns	2.4+
Turns to cycle a 10-card deck	2 turns

In-Game Thought Process

1. Know your coin density — total coins ÷ deck size at all times
2. Estimate key card draw probability — copies ÷ deck size × 5
3. Judge estate removal priority — early game: remove first; mid game: weigh against economy
4. Project coronation timing — per-turn probability × remaining turns
5. Calculate succession point expectations — remaining turns × expected points per turn

Mathematical thinking provides a framework for making decisions when you're uncertain what to buy. Combining intuition with numbers will steadily improve your win rate in Heart of Crown.