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Category Statistics & Probability

Total Tiles in Pool (N) How many tiles remain in the bag/pool (1–500)

Target Tiles in Pool (K) Count of the specific tile type you want (0–N)

Tiles Drawn (n) Number of tiles drawn at once (1–N)

Desired Target Count (k) Exactly how many target tiles you want drawn (0–min(K,n))

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About

Tile-based games like Scrabble, Mahjong, Azul, and Rummikub demand probabilistic reasoning. Every draw from a finite pool shifts the odds for every subsequent draw. The governing model is the hypergeometric distribution, which computes exact probabilities for sampling without replacement. Unlike the binomial model, it accounts for the shrinking population after each draw. Miscalculating these odds leads to suboptimal strategy - holding tiles too long, misjudging defensive plays, or underestimating an opponent's likely hand. This tool computes P(X = k) and cumulative probabilities using exact combinatorial arithmetic, not Monte Carlo approximation. It handles pools up to 500 tiles with no rounding until final display.

Limitations: the model assumes uniform random drawing (no peeking, no weighted bags). It does not account for tile-trading mechanics or multi-stage draws with intermediate decisions. Pro tip: after each draw, update the remaining pool count and re-check - conditional probability shifts faster than intuition suggests, especially when fewer than 20 tiles remain.

Formulas

The probability of drawing exactly k target tiles follows the hypergeometric probability mass function:

P(X = k) = C(K, k) ⋅ C(N − K, n − k)C(N, n)

The binomial coefficient C(a, b) counts combinations:

C(a, b) = a!b! ⋅ (a − b)!

Where N = total tiles in the pool, K = number of target tiles in the pool, n = number of tiles drawn, k = desired count of target tiles in the draw. The cumulative probability of drawing at least one target tile is:

P(X ≥ 1) = 1 − P(X = 0) = 1 − C(N − K, n)C(N, n)

The expected number of target tiles drawn is:

E(X) = n ⋅ KN

Variance of the distribution:

Var(X) = n ⋅ KN ⋅ N − KN ⋅ N − nN − 1

Reference Data

Game	Total Tiles	Distinct Types	Tiles Per Draw	Notable Probability Scenario
Scrabble (English)	100	27	7	P(no vowel in opening hand) ≈ 0.7%
Mahjong (Standard)	144	42	13	P(all suits in hand) ≈ 98.5%
Azul	100	5	4 (factory)	P(monochrome factory) ≈ 0.084%
Rummikub	106	53	14	P(both jokers in hand) ≈ 1.7%
Bananagrams	144	26	21	P(no S tile) ≈ 18.6%
Qwirkle	108	36	6	P(matching color set) ≈ 0.51%
Dominoes (Double-6)	28	28	7	P(holding double-6) = 25%
Dominoes (Double-9)	55	55	7	P(holding double-9) ≈ 12.7%
Carcassonne	72	24	1	P(specific tile next) ≤ 12.5%
Codenames	25	4	1	P(assassin on guess) = 4%
Upwords	100	26	7	P(all consonants) ≈ 2.1%
Letter Jam	64	26	1	P(vowel) ≈ 39%
Sagrada	90	5	4	P(all same color pull) ≈ 0.11%
Splendor	40 (gems)	5	3	P(3 same color available) varies by state
Kingdomino	48	6	4	P(crown tile in round) ≈ 58%

Frequently Asked Questions

Each tile removed reduces both N (total pool) and potentially K (target count). The hypergeometric model inherently handles this - simply update the pool parameters after each draw. The shift is nonlinear: removing 5 tiles from a pool of 100 has a smaller proportional effect than removing 5 from a pool of 20. Always recalculate after each draw event rather than relying on initial odds.

The model assumes: (1) tiles are drawn uniformly at random without replacement, (2) the pool composition is exactly known, and (3) no information leaks between draws (e.g., seeing partial faces). If tiles are sticky, heavier, or mechanically biased, the uniform assumption fails. For very large pools where K/N is stable, the binomial approximation works - but for typical board games with N < 200, exact hypergeometric calculation is both feasible and necessary.

The binomial distribution models draws with replacement (each draw is independent). As N → ∞ while K/N remains constant, the hypergeometric converges to the binomial. In practice, if the draw size n is less than 5% of N, the binomial approximation introduces under 1% relative error. For small pools (board games), this threshold is rarely met.

This tool computes odds for a single target category per calculation. For joint probabilities (e.g., drawing at least 1 red AND at least 1 blue), you need the multivariate hypergeometric distribution, which multiplies combinatorial terms across categories. As a workaround, compute each category independently and note that the joint probability is bounded above by the minimum of the individual probabilities.

A result of 0% means the scenario is combinatorially impossible - for example, requesting k = 5 target tiles when only K = 3 exist in the pool. A result of 100% means it is guaranteed - for example, drawing n = 10 tiles when only K = 10 non-target tiles exist and n > N − K. These are not rounding artifacts; they reflect hard combinatorial constraints.

Yes. Set N to the remaining tile count (starting at 100 for English Scrabble), K to the count of a specific letter remaining (e.g., 12 for the letter E at game start), and n to your rack size (7). After each turn, subtract all placed and drawn tiles from both N and K as appropriate, then recalculate.