Problem of points

The problem of points is a foundational problem in probability theory that addresses how to fairly divide the stakes in a game of chance when the game is interrupted before completion, ensuring each player receives a share proportional to their probability of winning from the current position.^[1]^[2]^[3] Originating in the late 15th century, the problem was first documented by the Italian mathematician Fra Luca Pacioli in his 1494 treatise Summa de arithmetica, geometria, proportioni et proportionalità, where it appeared in the context of gambling disputes.^[1] Gerolamo Cardano, in his 1525 work Liber de ludo aleae, provided an early but flawed solution based on proportional division of remaining rounds, which did not fully account for probabilistic outcomes.^[1] The issue gained renewed attention in 17th-century France amid the intellectual circles of scholars like those in the Mersenne group, where gambling problems were common topics.^[3] In 1654, the problem was formally resolved through an exchange of letters between Blaise Pascal and Pierre de Fermat, prompted by the French gambler Chevalier de Méré, marking a pivotal moment in the birth of modern probability theory.^[1]^[2] Fermat proposed a combinatorial method, enumerating all possible future outcomes to determine each player's share—for instance, in a game where one player needs one more point and the other needs three, with equal chances per round, the shares are 7/8 and 1/8 of the stake, respectively.^[2]^[3] Pascal developed a more efficient approach using expected value and his arithmetic triangle (now known as Pascal's triangle), which simplified calculations for larger games; for example, in a scenario with two players each starting with 32 pistoles, one having two points and the other one, the division would be 48 and 16 pistoles.^[1]^[2] Their methods converged for two-player games but highlighted differences for multiple players, where Pascal adjusted for early termination conditions.^[2] The resolution of the problem of points not only established formal rules for probabilistic reasoning but also influenced subsequent developments in mathematics, including binomial distributions and decision theory under uncertainty.^[3] It remains a cornerstone example in teaching probability, illustrating the shift from intuitive fairness to rigorous quantification.^[1]

Historical Background

Origins in Gambling Games

The problem of points originated in the gambling practices of Renaissance Europe, particularly in Italy during the 15th and 16th centuries, where players wagered stakes on games of chance that required accumulating a fixed number of points or rounds to win. These games, often involving dice, cards, or other competitive activities, frequently ended prematurely due to time constraints, fatigue, or external interruptions, leaving unresolved the fair distribution of the pooled stakes. In such scenarios, participants sought a method to divide the pot proportionally to their progress, reflecting the era's growing interest in equitable risk-sharing amid a burgeoning culture of betting among nobility and merchants.^[4]^[5] Italian gambling culture flourished in this period, with popular pastimes including card games like primiera—a vying game where players scored points based on card combinations and could play multiple hands—and bassetta, a later banking game involving bets on card draws that emphasized chance over skill. Other activities, such as dice rolling or even non-card games like pallone (a ball game) and crossbow shooting contests, similarly involved point accumulation to determine victors and stake winners. These pursuits were widespread in urban centers like Venice and Florence, despite periodic bans, as they intertwined entertainment, social status, and financial speculation.^[6]^[4] The issue gained mathematical attention in 1494 through Luca Pacioli's influential treatise Summa de arithmetica, geometria, proportioni et proportionalita, the first comprehensive printed work on arithmetic in Europe. In its section on games of chance, Pacioli informally addressed stake division in interrupted contests, using examples like a ball game requiring 60 points (interrupted at 50 for one player and 20 for the other) and a crossbow match to six goals involving three players (stopped with scores of 4, 3, and 2). He advocated proportional allocation based on points earned relative to the game's structure, marking an early attempt to systematize fairness without a probabilistic framework.^[7]^[4] This approach highlighted the core fairness dilemma: while players intuitively desired divisions reflecting their achieved progress, the absence of a reliable method to account for remaining uncertainties often led to disputes, as simple ratios ignored potential future outcomes. No systematic resolution existed at the time, setting the stage for later advancements by figures like Blaise Pascal and Pierre de Fermat in the 17th century.^[7]^[5]

Early Proposed Solutions

One of the earliest documented attempts to address the problem of points came from Luca Pacioli in his 1494 treatise Summa de arithmetica, geometrica, proportioni et proportionalità, where he suggested dividing the stakes proportionally to the points already won relative to the maximum possible in the game, as in the ball game example interrupted at 50-20 (out of 60 needed), yielding shares of approximately 7:3 ducats.^[4] This approach, however, was criticized for its failure to account for the remaining rounds and the uncertainty of future outcomes, leading to intuitively unfair results, such as awarding nearly the entire stake after just a few points if the game was interrupted early.^[4] In the mid-16th century, Niccolò Tartaglia critiqued Pacioli's proportional division in his Trattato generale di numerie et misure (1556), arguing that it ignored the points needed to win and proposed instead a method based on the ratio of the lead to the total game length to better approximate the chances of winning the remaining points.^[8] For example, in a first-to-10 game interrupted at 7-4, Pacioli's method yields a 7/11 to 4/11 split of the stakes, while Tartaglia adjusted for the remaining opportunities without full enumeration, aiming to reflect differential advantages.^[5] Yet Tartaglia's heuristic was flawed, as it produced the same division for a 65-55 lead as for a 99-89 lead in a first-to-100 game, overvaluing equivalent leads regardless of how close the game was to conclusion.^[8] Gerolamo Cardano discussed gambling interruptions in his Liber de ludo aleae (written around 1564, published posthumously in 1663) and earlier Practica arithmetica (1539), offering rough heuristics like dividing stakes based on the sum of an arithmetic progression of points still needed by each player, rather than a general probabilistic rule.^[4] In one case, for a game to 10 points interrupted at 9-7, Cardano suggested a 6:1 ratio derived from summing the series 1+2+3 for the trailing player's needs, allocating most of the stake to the leader.^[4] This method, while attempting to incorporate remaining efforts, lacked consistency and failed to systematically evaluate winning probabilities, often resulting in arbitrary outcomes.^[5] Other Italian mathematicians of the period and those referenced in contemporary treatises employed ad hoc rules like scaling divisions by the lead relative to total possible rounds or fixed proportions based on past performance, but these exhibited inconsistencies, such as overvaluing large leads in games with few rounds left while undervaluing them in longer contests.^[4] These intuitive but erroneous approaches highlighted the need for a more rigorous framework, eventually prompted by the Chevalier de Méré's query to Blaise Pascal in 1654.^[9]

Problem Formulation

Classic Statement

The problem of points, also known as the division of stakes, concerns the fair allocation of a total stake S between two players, A and B, in a game of chance that is interrupted before completion.^[10] The game consists of a series of independent rounds, each won by A or B with equal probability $1/2, and continues until one player reaches a predetermined number n of points, at which point that player claims the entire stake.^[10] The interruption occurs when A has accumulated a points and B has b points, with a < n and b < n.^[2] Under the key assumptions of independence across rounds and no skill differential between players, the focus shifts to the remaining points required: r = n - a for A and s = n - b for B.^[10] The fairness criterion mandates that the stake be divided in proportion to each player's expected winnings if the game were to continue under these conditions, which corresponds to the probability that A would ultimately reach n points before B does (with B receiving the complementary share).^[10] Boundary cases clarify the extremes: if one player has already reached n points (e.g., r = 0 or s = 0), that player receives the full stake S, as the game is effectively over.^[11] Similarly, if the players have equal points (a = b < n, so r = s), the stake is divided equally due to symmetry in their probabilities of victory.^[10] This problem was posed in its classic form by the gambler Chevalier de Méré to Blaise Pascal around 1654.^[2]

Illustrative Examples

To illustrate the problem of points, consider simple scenarios where two equally skilled players, A and B, contribute equally to a total stake and alternate fair trials (each with equal chance of winning a point) until one reaches a predetermined number of points. These examples mirror the queries posed by Chevalier de Méré to Blaise Pascal in 1654.^[10]^[12] A classic case arises in a game to 3 points interrupted when A leads 2-1. A simple proportional division of the stake based on current points—awarding A two-thirds and B one-third—might seem intuitive, as it reflects the accumulated progress, but it raises concerns about fairness given the remaining points needed: A requires just one more win, while B needs two.^[10]^[12] For simplicity, assume the total pot S = 1; the 2/3 : 1/3 split thus yields A = 2/3 and B = 1/3, yet this approach overlooks the asymmetric paths to victory and feels inadequate without deeper analysis.^[10] Another scenario involves a longer game to 10 points, halted at 8-7 with A ahead. Here, A's slim one-point lead late in the contest intuitively suggests a strong likelihood of A securing the win, as A needs only two more points while B requires three; a proportional split of 8/15 : 7/15 (approximately 0.533 : 0.467 for S = 1) understates this advantage and highlights how accumulated points alone fail to capture the game's momentum.^[12]^[10] In contrast, a symmetric interruption at 5-5 in a game to 10 points justifies an equal split of the stake (S = 1 divided as 0.5 : 0.5), as both players face identical remaining requirements and equal chances. This even division aligns with intuition in balanced cases but underscores the problem's nuance in asymmetric situations, where proportional methods diverge from expected fairness.^[10]^[12]

Pascal and Fermat's Contributions

Fermat's Enumeration Approach

In 1654, Pierre de Fermat proposed a combinatorial solution to the problem of points through the exhaustive enumeration of possible outcomes in a hypothetical completion of the game.^[13] Suppose player A requires r more points to win and player B requires s more points, with each round won by A or B with equal probability $1/2. Fermat's method imagines playing exactly r + s - 1 additional rounds, the minimum number sufficient to ensure one player reaches their required points.^[2] This produces $2^{r+s-1} equally likely sequences of outcomes, as each round has two possible results.^[13] The sequences favorable to A are those in which A wins at least r of these r + s - 1 rounds, ensuring that A reaches r points before B can accumulate s points in the sequence.^[2] Equivalently, these are the outcomes where B wins fewer than s rounds. The number of such favorable sequences is given by the sum of binomial coefficients:

\sum_{k=r}^{r+s-1} \binom{r+s-1}{k}

Thus, A's share of the total stake S is

\frac{1}{S} \sum_{k=r}^{r+s-1} \binom{r+s-1}{k} \cdot \frac{S}{2^{r+s-1}} = S \cdot \frac{\sum_{k=r}^{r+s-1} \binom{r+s-1}{k}}{2^{r+s-1}}.

^[13] This approach directly computes the probability by counting paths that avoid B reaching s wins first.^[2] Fermat outlined this enumeration method in his letter to Blaise Pascal dated September 25, 1654, describing the hypothetical rounds as a "fiction… [that] serves only to make the rule easy and… make all the chances equal."^[13] He noted that while the method assumes equal probabilities per round, it extends to unequal probabilities by assigning weights proportional to the odds of each outcome rather than treating all sequences as equiprobable.^[2] In contrast to Pascal's recursive method based on expected values, Fermat's enumeration provides a forward-looking count of complete game paths.^[13]

Pascal's Expected Value Method

Blaise Pascal developed a method to resolve the problem of points by defining the fair value of a player's position in terms of expected winnings, assuming fair dice or coins where each round gives a 1/2 probability of advancing for either player. In this framework, if the game is interrupted with player A needing r more points to win and player B needing s more points, the stake S is divided proportionally to each player's expected share, computed by considering the potential outcomes of continuing play. This approach, outlined in Pascal's correspondence with Pierre de Fermat, emphasizes recursive calculation to determine these expectations without enumerating all possible full-game outcomes.^[13] Pascal formalized the recursive relation for the expected value V(r, s), representing player A's fair share of the stake when A needs r points and B needs s points. The formula is given by

V(r, s) = \frac{1}{2} V(r-1, s) + \frac{1}{2} V(r, s-1),

with boundary conditions V(0, s) = S (A wins immediately) for s > 0, and V(r, 0) = 0 (B has already won) for r > 0. This recursion models the expected value after one more round: with equal probability, either A advances (reducing to V(r-1, s)) or B advances (to V(r, s-1)). Computations proceed backward from the boundaries, filling a table of values step by step until reaching V(r, s).^[13] To compute these values efficiently, Pascal employed tables constructed iteratively from the end states, which naturally align with the structure of his arithmetical triangle (now known as Pascal's triangle). Each entry in the table corresponds to binomial coefficients, allowing the recursion to build cumulatively; for instance, the values propagate like paths in the triangle, yielding results equivalent to sums of binomial probabilities. This tabular method facilitates step-by-step evaluation for specific r and s, avoiding exhaustive listings.^[14] The recursive process leads to a closed-form expression for A's share: V(r, s) = S \cdot \sum_{k=0}^{s-1} \binom{r+s-1}{k} / 2^{r+s-1}, where the sum represents the probability that A wins at least r of the next r+s-1 rounds before B reaches s successes. This derivation arises directly from expanding the recursion, interpreting the terms as the probabilities of favorable paths weighted by the stake.^[14] Pascal first detailed this expected value method in his July 29, 1654, letter to Fermat, where he applied it to specific cases like a 2:1 interruption in a game to 3 points. He later connected it explicitly to binomial coefficients in his Traité du triangle arithmétique (1665), demonstrating how the triangle's entries provide the necessary combinatorial counts for dividing stakes in interrupted games.^[13]^[14]

Mathematical Analysis

Binomial Coefficient Solutions

The binomial coefficient approach provides a closed-form solution to the problem of points by interpreting the probability of one player winning as the cumulative distribution of a binomial random variable, unifying Fermat's combinatorial enumeration with Pascal's recursive expected value method under modern probability notation.^[10] In the classic fair game where each point is won by player A or B with equal probability \frac{1}{2}, and A requires r more points to win while B requires s more points, Fermat's method of imagining the game continues for exactly r + s - 1 additional points yields the probability that A wins as the chance that B scores fewer than s points in those trials. This is given by

P(A \ wins) = \sum_{k=0}^{s-1} \binom{r+s-1}{k} \left( \frac{1}{2} \right)^{r+s-1},

where k represents the number of points B would score.^[13] Equivalently, since the roles are symmetric in the fair case, this is the probability that A scores at least r points in r + s - 1 trials:

P(A \ wins) = \sum_{j=r}^{r+s-1} \binom{r+s-1}{j} \left( \frac{1}{2} \right)^{r+s-1}.

^[10] This formula arises from the binomial theorem, which states that the total probability over all outcomes sums to 1:

\sum_{k=0}^{r+s-1} \binom{r+s-1}{k} \left( \frac{1}{2} \right)^{r+s-1} = 1,

so P(A \ wins) = 1 - P(B \ wins), where P(B \ wins) = \sum_{k=0}^{r-1} \binom{r+s-1}{k} \left( \frac{1}{2} \right)^{r+s-1} from B's perspective of needing at least s points.^[13] The binomial coefficients \binom{r+s-1}{k} count the number of sequences with exactly k successes for one player, aligning directly with Fermat's enumeration of equally likely outcomes.^[10] To verify equivalence with Pascal's method, note that the recursive expected value computation satisfies the boundary conditions and recurrence of the binomial probabilities; for instance, the partial sums from Pascal's triangle (the rows of binomial coefficients) match the enumerated counts in Fermat's approach, confirming both yield the same division of stakes proportional to these probabilities.^[13] A concrete illustration is the case where the game is to 3 points, A has 2 points (needs r = 1 more), and B has 1 point (needs s = 2 more). Here, r + s - 1 = 2, so

P(A \ wins) = \sum_{k=0}^{1} \binom{2}{k} \left( \frac{1}{2} \right)^{2} = \frac{\binom{2}{0} + \binom{2}{1}}{4} = \frac{1 + 2}{4} = \frac{3}{4}.

Thus, if the total stake is 4 units, A receives 3 units and B receives 1 unit.^[10] This matches Fermat's listing of the 4 possible outcomes of 2 trials (AA, AB, BA, BB), where A wins in all but BB, and Pascal's recursion: with probability \frac{1}{2} A wins the next point outright (full stake), or with \frac{1}{2} it ties at 2-2 and stakes split equally thereafter.^[13] For unequal probabilities, where A wins each point with probability p and B with q = 1 - p, the formula generalizes to

P(A \ wins) = \sum_{j=r}^{r+s-1} \binom{r+s-1}{j} p^j q^{r+s-1-j},

weighting the sequences by their actual probabilities rather than assuming uniformity.^[10] This extension, while beyond the original fair-game correspondence, follows naturally from the binomial model and was later formalized in early probability texts.^[13]

Generalizations and Formulas

The problem of points generalizes naturally to cases where the players have unequal probabilities of winning each round, denoted by p for player A and q = 1 - p for player B, with p \neq q. In this setting, suppose A needs r more points to win and B needs s more points. The probability that A wins is the probability that the number of failures (B's points) before A's r-th success is less than s, which follows the negative binomial distribution. This probability is given by

P(A \ wins) = \sum_{j=0}^{s-1} \binom{r + j - 1}{j} p^r q^j.

Equivalently, it can be expressed as the cumulative sum \sum_{k=r}^{r+s-1} \binom{r + s - 1}{k} p^k q^{r + s - 1 - k}.^[15] An extension to infinite stakes occurs when one player requires an arbitrarily large number of additional points, effectively modeling the probability of eventual ruin in an unbounded game. This leads to the gambler's ruin framework, where the finite interruption of the original problem corresponds to a gambler starting with capital i facing total capital N, but the analysis remains limited to scenarios with finite horizons rather than true infinity. For unequal probabilities, the probability that the gambler reaches N before ruin (0) is \frac{1 - (q/p)^i}{1 - (q/p)^N} when p \neq q.^[16] For multi-player variants or asymmetric targets (e.g., more than two players or different starting deficits to the winning threshold n), the probabilities satisfy adjusted recursive relations derived from Markov chain transitions. In the case of k players with initial "chips" c_1, \dots, c_k and first-to-n rule, the win probabilities can be computed via solving a system of linear equations, generalizing the two-player recursion to higher dimensions. Computationally, these generalizations are often solved using dynamic programming algorithms, where the value function for the probability at each state (remaining points needed) is filled recursively, tracing roots to Pascal's original recursive method for the equal-probability case. Modern implementations leverage matrix methods or iterative solvers for efficiency in multi-player settings.

Influence and Legacy

Development of Probability Theory

The problem of points, resolved through the 1654 correspondence between Blaise Pascal and Pierre de Fermat, provided a crucial catalyst for the formalization of probability theory in the mid-17th century. Christiaan Huygens, building directly on Pascal's ideas, published De Ratiociniis in Ludo Aleae in 1657, the first treatise dedicated to probability. In it, Huygens systematized the concept of expected value, deriving fair prices for games of chance and treating the problem of points as a central example to illustrate equitable stake division in interrupted contests. This work extended Pascal's recursive method of backward induction, emphasizing symmetry in outcomes and laying the groundwork for analyzing risks in gambling scenarios.^[17]^[9] The dissemination of these ideas accelerated through key publications in the latter half of the 17th century. Pascal's Traité du triangle arithmétique, written around 1654 but published posthumously in 1665, popularized binomial coefficients as a tool for solving the problem combinatorially, offering a general framework for counting favorable outcomes in series of trials. Fermat's contributions, meanwhile, circulated primarily through private letters rather than formal publication, influencing contemporaries like Huygens and ensuring the problem's methods reached broader mathematical circles across Europe. These texts shifted the focus from ad hoc resolutions to structured mathematical reasoning, enabling precise calculations for incomplete games.^[17]^[9] In the ensuing decades, the problem's solutions permeated 17th-century mathematical correspondence, notably impacting the Bernoulli family and other scholars who expanded probability beyond single events to long-run frequencies. Jakob Bernoulli, in his posthumously published Ars Conjectandi (1713), drew on these foundations to develop the law of large numbers, proving that empirical frequencies converge to theoretical probabilities over repeated trials—a direct evolution from the combinatorial approaches to the problem of points. This correspondence network, including figures like Gottfried Wilhelm Leibniz, fostered a conceptual shift toward probability as a measure of long-term expectation rather than mere chance.^[17]^[9] Ultimately, the problem of points transformed the mathematics of gambling from intuitive heuristics to a rigorous discipline, providing essential tools for emerging fields like insurance and annuities. By quantifying uncertainty through expected values and binomial expansions, it enabled the modeling of life contingencies and risk pooling, as later applied by Abraham de Moivre in his 1725 annuity tables based on mortality probabilities. This foundational rigor influenced actuarial practices, establishing probability as a practical framework for financial decision-making under uncertainty.^[17]^[9] The gambler's ruin problem represents a key 17th-century extension of the problem of points, formulated by Christiaan Huygens in his 1657 treatise De ratiociniis in ludo aleae, where the game continues indefinitely until one player loses all stakes, assuming finite initial capital for both participants but equal probabilities of winning each round.^[18] This variant employs recursive probability calculations akin to those in the original problem to determine the likelihood of eventual ruin. For unequal win probabilities p and q = 1 - p, with the first player starting with a units and the opponent with b units, the probability of the first player's ruin is given by

\frac{\left( \frac{q}{p} \right)^a \left[ 1 - \left( \frac{q}{p} \right)^b \right]}{1 - \left( \frac{q}{p} \right)^{a+b}}

when p \neq q, reducing to b / (a + b) in the fair case p = q = 1/2.^[19]^[20] The ballot theorem provides an 19th-century combinatorial generalization building on the enumerative techniques from the problem of points, quantifying the number of favorable paths where one player maintains a strict lead throughout a sequence of fair trials, such as vote counts. First articulated by Joseph Bertrand in 1887, it states that if candidate A receives a votes and B receives b votes with a > b, then the probability that A is always ahead during the count is (a - b) / (a + b).^[21] This result derives directly from the path-counting methods used in early solutions to the points problem, linking it to broader random walk theory. In modern probability, the problem of points informs applications in decision theory, particularly optimal stopping rules for interrupted contests, where stakes must be divided based on current states to maximize expected utility. It also appears in Bayesian updating frameworks for assessing game states, such as estimating posterior probabilities of victory from partial outcomes in sequential trials.^[22] The problem's foundational role in combinatorial probability was recognized in the 20th century, with William Feller's An Introduction to Probability Theory and Its Applications (1950) highlighting it as the origin of systematic enumerative methods in the field.