Nothing is more misleading than a poorly constructed inductive argument. After all, are we not warned? "Just because a coin has come up "heads" five times in a row does not mean it is less likely to come up "heads" on the sixth toss!" the teacher cautions us, before adding, with each syllable sharply enunciated: "The events are in-de-pen-dent." Conversely, how can we fail to see a pattern when fate persists? The 20th-century mathematician Bertrand Russell (see our feature in Tangente 206, 2022) cited the example of a Christmas turkey: treated generously every morning, does it not come to trust the hand that feeds it, even though that same hand will wring its neck on Christmas Eve?
Whom, then, should we believe? The teacher, Russell, or… Laplace? All three, because everything depends on the assumptions! In particular, the Bayesian approach sheds light on Laplace's calculation, which yielded the probability of the Sun rising again the next day as 0.99999945.
"Independent" tosses
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Laplace laid the foundations for his reasoning as early as 1774 in his treatise Mémoires sur la probabilité des causes par les évènements, and illustrated it forty years later by considering the succession of days in his Essai philosophique sur les probabilités.
Let us begin with a textbook example: a one-euro coin. First suppose that it is fair, and toss it several times. The tosses take place under conditions entirely unrelated to one another: in everyday language, they are "independent." We may therefore assume that the successive events "heads" and "tails" are independent in the mathematical sense. The coin thus retains no memory of previous tosses.
