12.2 Pascal, Fermat, and the foundations of probability theory

Early Probability Pioneers

Probability theory didn't emerge from abstract mathematics. It grew out of a very practical question: how should gamblers split the pot when a game gets interrupted? In 1654, Blaise Pascal and Pierre de Fermat exchanged a series of letters that tackled this question head-on, and their correspondence became one of the founding documents of probability theory.

Pascal and Fermat's Contributions

Blaise Pascal (1623–1662) was a French mathematician, physicist, and philosopher already known for work on geometry and mechanical calculators. Pierre de Fermat (1601–1665), a lawyer by profession, was one of the most talented mathematicians of his era, famous for contributions to number theory.

Their collaboration on probability began not in a university but through a gambling dispute. Antoine Gombaud, known as the Chevalier de Méré, posed questions to Pascal about fair division of stakes and dice odds. Pascal then wrote to Fermat, and the two developed competing but complementary approaches to these problems. Pascal favored recursive reasoning and what we'd now call combinatorial arguments, while Fermat leaned on systematic enumeration of all possible outcomes.

What made their exchange groundbreaking was the shift from intuition to rigorous calculation. Before Pascal and Fermat, gambling odds were understood through experience and rough rules of thumb. After their correspondence, chance could be quantified precisely.

The Problem of Points

The problem of points is the specific question that sparked their correspondence: if two players are in the middle of a game of chance and must stop early, how should they divide the stakes fairly based on each player's current score?

For example, suppose two players each bet 32 pistoles on a game where the first to win 3 rounds takes the pot. If the game stops when Player A has won 2 rounds and Player B has won 1, how much should each receive?

Here's how Pascal and Fermat reasoned through it:

1. Identify how many more rounds could possibly be played (in this case, at most 2 more).
2. List every possible outcome of those remaining rounds (AA, AB, BA, BB).
3. Determine which outcomes result in a win for each player. Player A wins in 3 of the 4 scenarios (AA, AB, BA); Player B wins only if BB occurs.
4. Divide the stakes proportionally: Player A should receive $\frac{3}{4}$ of the pot, and Player B should receive $\frac{1}{4}$.

This approach was revolutionary because it based the division on future possibilities, not just the current score. It introduced the concept of expected value into mathematical thinking, and it laid the groundwork for probability calculations used centuries later in finance, insurance, and decision-making.

Foundations of Probability Theory

Probability Axioms and Basic Concepts

While Pascal and Fermat worked with probability informally, the formal axioms weren't established until Andrey Kolmogorov published them in 1933. Still, these axioms formalize the intuitions that Pascal and Fermat relied on, so understanding them helps you see where the early ideas led.

The three axioms are:

• Axiom 1 (Non-negativity): The probability of any event is a non-negative real number. $P(A) \geq 0$ for any event $A$.
• Axiom 2 (Normalization): The probability of the entire sample space (the set of all possible outcomes) equals 1. $P(S) = 1$.
• Axiom 3 (Additivity): For mutually exclusive events (events that can't happen at the same time), the probability of either one occurring equals the sum of their individual probabilities. $P(A \cup B) = P(A) + P(B)$ when $A$ and $B$ are mutually exclusive.

Every rule you encounter in probability can be derived from these three axioms. They provide the logical foundation that turns probability from a collection of gambling tricks into a rigorous branch of mathematics.

Expected Value and Its Applications

Expected value is the weighted average of all possible outcomes of a random event, where each outcome is weighted by its probability. It tells you what result to expect on average over many repetitions.

The formula:

$E(X) = \sum_{i=1}^{n} x_i \cdot p(x_i)$

where $x_i$ represents each possible outcome and $p(x_i)$ is the probability of that outcome.

Consider a simple example from the era of Pascal and Fermat: a game where you roll a fair die and win 6 coins if you roll a six, but pay 1 coin otherwise. Your expected value per roll is:

$E(X) = 6 \cdot \frac{1}{6} + (-1) \cdot \frac{5}{6} = 1 - \frac{5}{6} = \frac{1}{6}$

On average, you gain about 0.17 coins per roll. This is exactly the kind of calculation that grew out of Pascal and Fermat's work on the problem of points.

Expected value became a core tool well beyond gambling. Insurance companies use it to set premiums by calculating the average payout they'll owe. Financial analysts use it to compare investment strategies under uncertainty. In decision theory, choosing the option with the highest expected value is a foundational strategy.

Combinatorics in Probability

Combinatorics is the mathematics of counting arrangements and selections. Pascal and Fermat relied heavily on combinatorial reasoning to enumerate possible outcomes in their probability arguments.

Two key concepts:

• Permutations count ordered arrangements. If you're selecting $r$ items from $n$ items and the order matters, the formula is:

$P(n,r) = \frac{n!}{(n-r)!}$

• Combinations count unordered selections. If order doesn't matter, the formula is:

$C(n,r) = \frac{n!}{r!(n-r)!}$

The difference matters. Choosing 3 cards from a deck where the order of selection matters (permutation) gives a much larger number than choosing 3 cards where you only care about which cards you got (combination). Pascal's work on combinations was especially influential, as it connected directly to his triangle and to the binomial coefficients used throughout probability.

Mathematical Tools and Concepts

Pascal's Triangle and Its Properties

Pascal's triangle is a triangular array of numbers where each entry is the sum of the two entries directly above it. The triangle starts with 1 at the top, and the edges are always 1.

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        1
       1 1
      1 2 1
     1 3 3 1
    1 4 6 4 1

The entry in row $n$, position $r$ (both counting from 0) equals $C(n,r)$. So row 4 gives you $C(4,0) = 1$, $C(4,1) = 4$, $C(4,2) = 6$, $C(4,3) = 4$, $C(4,4) = 1$.

This makes the triangle a quick reference for binomial coefficients. If you expand $(a + b)^4$, the coefficients are exactly 1, 4, 6, 4, 1. Pascal didn't invent this triangle (Chinese and Persian mathematicians studied it centuries earlier), but he was the first to systematically connect it to probability and combinatorics in his 1665 work Traité du triangle arithmétique.

The triangle also contains other patterns: the diagonals give natural numbers, triangular numbers, and tetrahedral numbers. The sum of each row equals a power of 2 (row $n$ sums to $2^n$).

Advanced Combinatorial Techniques

Beyond basic permutations and combinations, several techniques extend combinatorial reasoning to harder problems:

• Principle of Inclusion-Exclusion: When counting elements that belong to overlapping sets, you add the sizes of individual sets, subtract the sizes of pairwise intersections, add back triple intersections, and so on. This prevents double-counting.
• Generating functions: These encode a sequence of numbers as coefficients of a polynomial or power series. They turn counting problems into algebraic ones, making complex combinatorial questions more tractable.
• Recurrence relations: These define each term in a sequence based on previous terms. The Fibonacci sequence is a classic example. In combinatorics, recurrence relations often arise when a counting problem at size $n$ can be broken into smaller subproblems.

These tools developed gradually after Pascal and Fermat's era, but they grew directly from the combinatorial foundations that the two mathematicians established.

Applications of Expected Value

The concept of expected value that Pascal and Fermat developed for gambling problems proved remarkably versatile:

• Gambling: Casinos use expected value to ensure every game has a slight edge in their favor. A player might win in the short term, but the expected value guarantees the house profits over thousands of plays.
• Finance: Portfolio managers compare investments by calculating the expected return (weighted average of possible gains and losses). This allows systematic comparison of options with different risk profiles.
• Insurance: Companies estimate the expected payout for a policy by multiplying the probability of each type of claim by its cost. Premiums are set above this expected value to ensure profitability.
• Decision theory: When facing choices under uncertainty, expected value provides a rational framework for selecting the best option. Pascal himself applied this reasoning beyond mathematics in his famous "Pascal's Wager," an argument about belief in God framed in terms of expected outcomes.

The thread connecting a 1654 letter about an interrupted dice game to modern risk analysis is remarkably direct. Pascal and Fermat gave us the tools to reason precisely about uncertainty, and those tools have only grown more essential since.