How Probability Was Born: Pascal, Fermat and a Gambling Problem

Today probability powers weather forecasts, insurance, medicine and artificial intelligence, yet the subject began with a very ordinary question: how should two gamblers split the money if their game is interrupted? In 1654 two French mathematicians, Blaise Pascal and Pierre de Fermat, answered it in a series of letters, and in doing so they created a new branch of mathematics.

Before 1654: Guesses and Near Misses

People have gambled with dice and bones for thousands of years, but for most of history nobody tried to measure chance mathematically. Outcomes were often seen as the will of fate or the gods.

A few earlier thinkers came close:

• Luca Pacioli, the Italian friar famous for describing double-entry bookkeeping, discussed the interrupted-game problem in 1494. His answer, splitting the stakes in proportion to games already won, turned out to be wrong.
• Gerolamo Cardano, an Italian physician and compulsive gambler, wrote Liber de Ludo Aleae (The Book on Games of Chance) around 1564. He understood the idea of counting equally likely outcomes, but the book was not published until 1663, long after his death, so it had little influence.
• Galileo Galilei, around the early 1600s, explained why rolling three dice gives a total of 10 slightly more often than a total of 9, by counting the 216 possible ordered outcomes.

What was missing was a general method that could handle unfinished games and future possibilities, not just finished counts.

The Chevalier de Méré's Puzzles

The spark came from Antoine Gombaud, a French writer and gambler known as the Chevalier de Méré. He brought two problems to Pascal, then a 31-year-old mathematician in Paris.

The first concerned dice. De Méré knew from experience that betting on getting at least one six in four rolls of a die was profitable. He reasoned that betting on at least one double six in 24 rolls of two dice should be equally good, because 24 is to 36 as 4 is to 6. But he seemed to lose money on the second bet. Why?

The second was the problem of points: two players each put money into a pot and agree that the first to win a certain number of rounds takes everything. If the game must stop early, how should the pot be divided fairly?

Pascal wrote to Fermat, a lawyer in Toulouse who did mathematics in his spare time. Their correspondence over the summer and autumn of 1654 is usually regarded as the birth of probability theory.

Worked Example 1: De Méré's Dice Paradox

The key trick is to find the probability of the opposite event, "no success at all," and subtract it from 1.

Bet A: at least one six in four rolls of one die.

The probability of not getting a six on one roll is \(\tfrac{5}{6}\). The rolls are independent, so the probability of no six in four rolls is

$$ \left(\frac{5}{6}\right)^4 = \frac{625}{1296} $$

Therefore

$$ P(\text{at least one six}) = 1 - \frac{625}{1296} = \frac{671}{1296} \approx 0.5177 $$

This is just above one half, so the bet wins slightly more often than it loses.

Bet B: at least one double six in 24 rolls of two dice.

Two dice have \(6 \times 6 = 36\) equally likely outcomes, and only one is a double six. So the probability of not getting a double six on one throw is \(\tfrac{35}{36}\). For 24 throws:

$$ \left(\frac{35}{36}\right)^{24} \approx 0.5086 $$

$$ P(\text{at least one double six}) = 1 - 0.5086 \approx 0.4914 $$

This is just below one half, so over many games the bet loses money. De Méré's proportional reasoning was wrong because probabilities of "at least one" do not grow in a straight line. His experience was correct, and it is impressive that he noticed a difference of under 2%.

Worked Example 2: The Problem of Points

Fermat's solution was to imagine the game continuing for the maximum number of rounds still needed, and to list every equally likely way those rounds could go, even if some would never actually be played.

Problem. Players A and B each stake 32 coins. The first to win 3 rounds takes all 64. Each round is a fair 50–50 contest. The game stops when A has 2 wins and B has 1. How should the 64 coins be divided?

Step 1: How many more rounds could be needed? A needs 1 more win and B needs 2. At most \(1 + 2 - 1 = 2\) more rounds would settle the game.

Step 2: List all outcomes of 2 rounds. They are AA, AB, BA, BB, each with probability \(\tfrac{1}{4}\).

Step 3: Decide the winner in each. A wins in AA, AB and BA, since A needs only one round. B wins only in BB.

Step 4: Compute the probabilities.

$$ P(A) = \frac{3}{4}, \qquad P(B) = \frac{1}{4} $$

Step 5: Divide the pot in that ratio. A receives \(\tfrac{3}{4} \times 64 = 48\) coins and B receives 16 coins.

Notice Pacioli's rule would have split the pot 2 : 1 according to wins so far, giving A only about 42.7 coins. That rule ignores what still needs to happen, which is exactly what matters.

A Harder Case

Suppose instead A needs 2 more wins and B needs 3 more. At most \(2 + 3 - 1 = 4\) more rounds are needed, giving \(2^4 = 16\) equally likely sequences. B wins only if A gets fewer than 2 of the 4 rounds:

• A wins 0 rounds: 1 sequence (BBBB).
• A wins exactly 1 round: 4 sequences.

So B wins in 5 sequences and A in the remaining 11:

$$ P(A) = \frac{11}{16}, \qquad P(B) = \frac{5}{16} $$

Pascal reached the same answers by a different route, working backwards from the end of the game and averaging the outcomes of each next round. He also connected these counts with his famous arithmetical triangle, now called Pascal's triangle, whose entries \(\binom{n}{k}\) count the ways to choose \(k\) wins out of \(n\) rounds. In the case above, \(1 + 4 = \binom{4}{0} + \binom{4}{1} = 5\).

From Gambling Tables to Science

The new ideas spread quickly.

• In 1657 the Dutch scientist Christiaan Huygens, having heard of the Paris work, published De Ratiociniis in Ludo Aleae, the first printed textbook on probability. He introduced the idea of expected value.
• Jacob Bernoulli's Ars Conjectandi (1713, published after his death) proved the first version of the law of large numbers: over many trials, the observed frequency settles near the true probability.
• Abraham de Moivre found the bell-shaped normal curve as an approximation to coin-tossing counts in the 1730s.
• Pierre-Simon Laplace's great treatise of 1812 applied probability to astronomy, errors in measurement and population statistics.
• In 1933 the Russian mathematician Andrey Kolmogorov finally placed probability on firm axioms, the foundations still taught today.

Did You Know?

• Pascal later used probability in a religious argument, now known as "Pascal's Wager," weighing the expected gains and losses of belief.
• Fermat and Pascal never met in person; their entire collaboration happened by letter.
• Insurance companies and casinos both rely on the law of large numbers: individual events are unpredictable, but averages over thousands of cases are very stable.

Key Takeaways

• Probability theory was born in 1654 from letters between Pascal and Fermat about gambling problems.
• De Méré's dice puzzle shows why "at least one" problems are best solved with \(1 - P(\text{none})\).
• The problem of points is solved by imagining all remaining rounds and dividing the pot by the probability of winning, not by games already won.
• Huygens, Bernoulli, de Moivre, Laplace and Kolmogorov turned these ideas into a full mathematical science.

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