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C E G L P V

Chance of Meeting a Target Number on One Die

p = \frac{d - t + 1}{d}

Games & RatingsStatisticsThe chance that a single fair die of d faces shows at least the target number t. Counting the faces that succeed is the whole derivation: there are d − t + 1 of them, and the plus one is the step everybody drops the first time.

Elo Expected Score

E_A = \frac{1}{1 + 10^{\,(R_B - R_A)/400}}

Games & RatingsStatisticsThe share of a point a player is expected to take against a given opponent, worked out from the difference between their two ratings alone. Half a point means an even match; the curve rises toward one as the gap grows and never quite reaches it. A 200-point advantage is worth about 0.76, which is the one number most players already carry in their heads.

Elo Rating Change After a Game

R' = R + K \, (S - E)

Games & RatingsStatisticsThe whole of an Elo update in one line: take the difference between what actually happened and what was expected, multiply by the K-factor, and add it to the old rating. Every point one player gains the other loses, so a closed pool's total rating never changes.

Expected Successes in a Dice Pool

E = \frac{n \, (d - t + 1)}{d}

Games & RatingsStatisticsThe average number of dice in a pool of n that meet or beat a target number t. It is the per-die chance multiplied by the number of dice — the binomial mean, arrived at without ever needing the binomial distribution, because expectation adds whether or not the dice are independent.

Expected Sum of Several Dice

E = \frac{n \, (d + 1)}{2}

Games & RatingsStatisticsThe long-run average total from rolling n identical dice of d faces each. One die averages the midpoint of its faces, and expectation adds, so n of them average n times that — which is why the answer lands on a half whenever the number of dice is odd.

Expected Value of an Exploding Die

E = \frac{d \, (d + 1)}{2 \, (d - 1)}

Games & RatingsStatisticsThe long-run average of a die that is rolled again and added whenever it lands on its highest face, with the rerolls themselves able to explode without limit. An ordinary six-sided die averages 3.5; the same die exploding averages 4.2, and the whole of that extra 0.7 comes from a geometric series that converges because each further explosion is six times rarer than the last.

Expected Value of the Higher of Two Dice

E_{\max} = \frac{(d + 1)(4d - 1)}{6d}

Games & RatingsStatisticsThe long-run average when two identical dice are rolled and only the larger is kept. On six-sided dice it is 161/36, about 4.47, against 3.5 for a single die — the extra is what an advantage on a roll is actually worth, and it is smaller than most people guess.

Expected Value of the Lower of Two Dice

E_{\min} = \frac{(d + 1)(2d + 1)}{6d}

Games & RatingsStatisticsThe long-run average when two identical dice are rolled and only the smaller is kept. On six-sided dice it is 91/36, about 2.53, against 3.5 for a single die. Added to the average of the higher die it gives exactly d + 1, and that identity is a complete proof that both formulas are correct.

Glicko Expected Score Against One Opponent

E = \frac{1}{1 + 10^{\,-g \, (r - r_j)/400}}

Games & RatingsStatisticsThe Elo curve with the rating difference first shrunk by the opponent's uncertainty. When the opponent's rating is exactly known the two agree; when it is not, this prediction sits closer to an even result, because a rating you cannot trust cannot support a confident forecast.

Glicko g Attenuation Factor

g(RD) = \frac{1}{\sqrt{1 + \dfrac{3 q^{2} RD^{2}}{\pi^{2}}}}, \qquad q = \frac{\ln 10}{400}

Games & RatingsStatisticsThe factor by which a rating difference is discounted when the opponent's own rating is uncertain. It is one when the opponent is perfectly known and falls toward zero as their rating deviation grows, which is Glicko's way of saying that a prediction can only be as confident as the weaker of the two ratings involved.

Glicko Rating Deviation Growth While Idle

RD = \sqrt{RD_0^{\,2} + c^{2} t}

Games & RatingsStatisticsHow the uncertainty attached to a rating grows during a layoff. Glickman's insight was that a rating is a claim with an error bar, and that the error bar widens whenever a player stops producing evidence — so a 1900 who last played eight years ago is a far weaker claim than a 1900 who played last week.

Glicko Rating Update (One Game)

r' = r + \frac{q \, g \, (s - E)}{\dfrac{1}{RD^{2}} + q^{2} g^{2} E (1 - E)}

Games & RatingsStatisticsGlicko's replacement for Elo's fixed K-factor: the step size is worked out from how well the rating was already known. A player with a wide rating deviation moves a long way on one result; a player whose rating rests on hundreds of games barely moves at all. The two terms in the denominator are exactly that trade — prior knowledge against the information one game carries.

Log5 Matchup Probability

p = \frac{p_A - p_A p_B}{p_A + p_B - 2 p_A p_B}

Games & RatingsStatisticsPredicts a head-to-head result from two win percentages against a common field. A .500 side beats a .400 side exactly 60% of the time, an even pair gives 0.500 whatever their shared strength, and the whole thing is the Bradley-Terry paired-comparison model wearing different clothes.

Performance Rating (Linear Approximation)

R_p = R_{avg} + \frac{400 \, (W - L)}{N}

Games & RatingsStatisticsWhat a player's results in one event were worth, expressed on the rating scale: the average rating of the opponents faced, adjusted by 400 points for every whole point of score above or below an even split. This is the linear form used for quick estimates, and it is an approximation of a definition that is otherwise iterative.

Pythagorean Expectation

W\% = \frac{RS^{\,k}}{RS^{\,k} + RA^{\,k}}

Games & RatingsStatisticsEstimates the win percentage a team deserved from the points it scored and the points it allowed, rather than from the games it happened to win. Bill James named it for its resemblance to the Pythagorean theorem when the exponent is two; the resemblance is the only thing the two have in common.

Variance of a Sum of Dice

\sigma^{2} = \frac{n \, (d^{2} - 1)}{12}

Games & RatingsStatisticsThe spread of the total from n identical dice of d faces. Because the dice are independent their variances simply add, which is why rolling several small dice gives a much tighter total than rolling one large die of the same average — the classic reason a designer chooses three six-sided dice over one twenty-sided one.