Against the Gods Ch. 2: 1654 — The Letters That Invented the Future

Investment Background

Chapter 1 explained why probability arrived so late. This chapter covers the moment it arrived.

In 1654 a French nobleman and gambler named the Chevalier de Méré posed a problem to the mathematician Blaise Pascal.

Pascal corresponded with Pierre de Fermat about it. Those letters are generally regarded as the birth of probability theory.

And the problem they solved has exactly the same structure as your retirement portfolio.

The Wall Street Translation

The Problem of Points

The problem runs as follows:

Two players each stake fifty gold coins, for a pot of one hundred. First to win five rounds takes everything.

The game is interrupted when player A has won four rounds and player B has won three.

How should the hundred coins be divided?

This had baffled mathematicians for two hundred years. Various answers had been proposed:

• Split it evenly? But A is clearly ahead; that is unfair.
• Divide by rounds won, four to three? The most popular answer at the time. It is still wrong.

Pascal and Fermat's insight: the division should be based not on what has already happened but on what would happen.

That sentence is the seed of all modern finance.

The Correct Solution

The key is asking: if the game continued, what would occur?

A needs one more round. B needs two. So at most two more rounds decide it.

All possible outcomes of two rounds — four of them, each equally likely:

Round one	Round two	Who ultimately wins
A wins	(no need to play)	A
B wins	A wins	A
B wins	B wins	B

Note the first row: if A wins round one, that is his fifth win and the game ends immediately. So across four equally likely paths, A wins in three.

The correct division is therefore three-quarters to A — seventy-five coins — and one-quarter to B, twenty-five.

Not four to three (roughly fifty-seven to forty-three). Three to one.

Why the Solution Was Revolutionary

Notice what they did: they assigned a numerical value to something that never happened and never will.

The game was interrupted. Those remaining rounds will never be played.

And Pascal and Fermat calculated their value anyway.

This is the birth of "expected value": an event that has not occurred can have a definite value computable right now.

Once that idea is accepted, all of modern finance becomes possible.

• Insurance: a policy's value is the expected value of future claims.
• Equity valuation: a company's value is the expected value of future cash flows.
• Option pricing: Scholes and Merton's formula computes the expected value of a future payoff.
• Your retirement plan: your portfolio's value in thirty years is an expected value.

All of it descends from those letters in 1654.

The Formula and a Trap

The calculation is extremely simple:

Expected value = each outcome's value multiplied by its probability, all summed.

And its most important use is revealing something counterintuitive: a high-probability bet can be a bad bet, and a low-probability bet can be a good one.

Two examples:

Bet A: 90% chance of winning $100, 10% chance of losing $1,000. Expected value = 0.9 × 100 minus 0.1 × 1,000 = negative $10. You win nine times, lose once, and lose money over time.

Bet B: 10% chance of winning $1,000, 90% chance of losing $50. Expected value = 0.1 × 1,000 minus 0.9 × 50 = positive $55. You lose nine times, win once, and make money over time.

Most people instinctively choose Bet A, because "you win more often."

This is exactly the content of "why a high win rate is a psychological trap" in Chapter 1 of Way of the Turtle, and the 90%-win-rate-with-negative-expectancy example in Chapter 1 of When Genius Failed.

What this book adds: the trap exists because expected value is itself counterintuitive — it took humanity two thousand years to invent.

Your intuition fails here not because you are foolish, but because human intuition evolved without this concept available.

An Important Qualification

Expected value has a severe limitation, and it is critical for retirement investors.

Expected value assumes you can repeat the bet enough times.

If a bet has positive expected value but failure removes you from the game, expected value is a misleading measure.

This is the entire content of Chapter 5 of When Genius Failed: right without staying power equals wrong.

A concrete example: a bet with a 99% chance of doubling your wealth and a 1% chance of taking it to zero.

The expected value is enormous.

And if you repeat it a hundred times, you are nearly certain to be ruined.

This is a mathematical fact, not a psychological problem. Chapter 6 of Antifragile in this library makes the same point through "the mathematics of Russian roulette."

So the correct usage is: use expected value to judge whether a bet is worth taking, and probability of ruin to judge how large it should be.

Two questions, two tools, not interchangeable.

Executable Trading Rules

1. For every decision, write out the outcomes and probabilities, then compute the expected value. The chapter's most practical line. Most people consider only the most likely outcome and ignore low-probability extremes. Writing it down forces you to face them.

2. Never evaluate a strategy by how often it wins. Use the average result per instance. A strategy winning 40% of the time at three-to-one beats one winning 80% at one-to-five.

3. Always ask "based on what will happen, not what already has." Pascal's core insight, whose direct investing application is ignoring your purchase price — the inventory frame in Chapter 4 of Trader Vic makes the same point from the other direction.

4. After computing expected value, check probability of ruin separately. Two questions: is the expected value positive? And if the worst case occurs, can I continue? Both must be yes.

5. Understand that your retirement calculator outputs an expected value. A Monte Carlo result of "85% success" means 15% of paths failed. Expected value will not tell you which path you are on.

Relevance to a Retirement Portfolio

This chapter explains the mathematical foundation of every tool on this site, and also their limits.

Pascal's insight made retirement planning possible: you can assign a number today to an outcome that will not resolve for thirty years, and decide on that basis. That was unimaginable before 1654.

But for a retirement investor, expected value carries one particularly important limitation.

Your retirement happens once.

Expected value describes the average result of long repetition. You will not repeat retirement thirty times. You retire once.

Which is why we consistently emphasize worst-case testing rather than expected outcomes alone:

Tool	What it gives you	What it does not
Expected return assumption	The average path	Which one you actually walk
Monte Carlo simulation	A distribution of success probability	Which percentile you land in
Worst-case testing	A lower bound	A probability

A retirement plan looking only at expected value treats a one-time, irreversible decision as a bet that can be repeated many times.

The correct approach: plan with expected value, test with the worst case, and absorb what neither covers with a cash buffer.

Chapter 3 covers the revolution's next step: when people began applying probability to the real world, they discovered the real world is not like dice.