Suppose two players, A and B, play a game where the first to reach three points wins, each round gives both players an equal chance to score, and every round is independent. The score reaches two to one, and the game is suddenly halted. Should the prize money be split evenly, or divided according to the score at that moment?
There is another way to divide it. First look at how likely A and B each are to win if the game were to continue. At this point A needs only one more point, while B still needs two. If A wins just one of the next two rounds, the game ends. The only way for B to come from behind is to win two rounds in a row, which has a probability of one quarter. Dividing by the chance of winning gives shares of three to one for A and B. This simplified example corresponds to the problem of points in the course. Outcomes that have not yet happened can also be brought into a decision made now, once the rules are clearly specified.
Counting from the problem of dividing the prize
Lesson 5 returns to the correspondence between Pascal and Fermat. When reading this history, one can begin by drawing the tree of outcomes for the game. If A wins the next round, the game ends immediately. Only if B wins the next round do the two need to play once more. After working through it, try changing one premise—for example, giving the two players different single-round win rates—and see how the division just described changes. The first step in any calculation is to state clearly what assumptions you have made about the game. Lesson 5, "The French Connection: Pascal and Fermat"
Risks at sea need someone to record them
When a ship will reach port, and what happened along the way, is hard to capture exhaustively with a simple outcome tree. Lesson 6 shifts the lens to population records, Halley's life tables, and the shipping news at Lloyd's coffee house. These dealt with very different subjects, yet each required that scattered events first be written down before there was any chance to compare them and form a judgment. Lesson 6, "The Historical Origins of Insurance"
Imagine yourself as the person reading the ledgers: seeing that several ships were delayed one after another, you can go on to check their departure times, the routes they took, and the weather they encountered. Only with this background can you tell which records are worth looking at together.
After computing the average outcome, ask who bears it
Lesson 7 moves from the St. Petersburg paradox to Daniel Bernoulli's view of utility. The expected payout of a game can be written into a formula, but participants must also consider the cost of entry and the effect that a possible loss would have on them. Imagine the same loss from a suspended voyage falling on an owner who operates a single ship, and on an owner who spreads operations across several ships: the weight of it may be very different. The utility discussed in the lesson is precisely about putting the person who bears the consequences back into the calculation. Lesson 7, "Considering Human Nature and Utility"
What to do when there are no comparable precedents
Lesson 14 brings readers to Keynes and Knight's discussion of uncertainty. For a game with known rules, outcomes and probabilities can be listed; but when a new situation lacks comparable records, historical numbers can offer only limited support. At such a point, what is worth writing down is the clues already in hand, the assumptions that most affect the conclusion, and what new information would change the judgment. Lesson 14, "Keynes versus Knight: The Nature of Uncertainty"
This general education course is based on Peter L. Bernstein's book of the same name and is suited to readers interested in economic history, probability and decision-making. Simplified Chinese course materials can be started from Lesson 5. First draw that unfinished match, then read the ledgers and shipping stories of the next lesson, and you will find that these two kinds of risk problems call for different kinds of evidence. Enter ECON001 "Against the Gods: The Remarkable Story of Risk"