I was reading this article, and I decided to use the “canonical example” of a “precocious newborn observing his first sunset” as a problem for my students (undergrads in an intro to inductive logic course). As noted here though: it doesn’t seem to lend itself too well to a basic analysis using Bayes’ rule.

Anyway, I came up with what’s below. Am I missing anything?

Deep in the forest of northern Mississippi, it was a dark and stormy night. A bolt of lightning struck the middle of a swamp; the water bubbled and sparks flew. Suddenly, an arm reached out of the muddy water and grabbed onto a log! The electrical current from the lightning must have rearranged the molecules in the swamp! Swampman---a creature looking just like an adult human male, but created only moments before by the fortuitous rearrangement of molecules in the swamp---was pulling himself onto the shore!

By the next evening, Swampman has cleaned himself up and gotten some clothes. All day he has observed his surroundings. He noticed that, when he crawled out of the swamp, it was dark, but then the sun rose, moved across the sky, and disappeared below the horizon. He wonders if this will happen again—will the sun rise again the next morning? He forms this hypothesis, which he refers to as H, the sun will rise tomorrow or, equivalently, since each day has a tomorrow, the sun rises every day. Since he has no other basis for setting the probability that H is true except to give each possibly---the sun will rise tomorrow or it won’t---equal weight, he makes P(H) = .50. He quickly sees that, given H---that is, if the sun rises everyday---then the probability of the sun rising on any particular morning is 1.0.

On the other hand, if H is false, the sun still might rise on any given morning, but the chance of that happening is less than 100 percent. He has already seen the sun rise once, though, so P(R|not H) has to be greater than 0. (R, which is the evidence, is the sun rose that morning.) Swampman mulls it over and decides to put the probability of the sun rising on any particular morning, given that H is false (i.e., the sun does not rise every day), at 50 percent also.

What are P(A), P(R|H), and P(R|not H)? What is the probability that H is correct, given that the sun rose Swampman’s first morning? That is, what is P(H|R)? The next morning, Swampman observes the sun rise again. Now, what is P(H|R)? How many times does Swampman have to observe the sun rising for P(H|R) to be higher than 95 percent?

Answer P(H) = .50, P(R|H) = 1.0, P(R|not H) = .50

$$P(H\mid R) = \frac{P(H) \times P(R\mid H)}{(P(H) \times P(R\mid H)) + ((P(\text{not } H) \times P(R\mid\text{not } H))} = \frac{.50 \times 1.0}{(.50 \times 1.0) + (.50 \times .50)} = .67$$

Subsequent iterations produce $P(H\mid R) = .80$, $.89$, $.94$, and $.97$.

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    $\begingroup$ This seems like a fine problem, but for MSE. $\endgroup$ – Mark Fantini Dec 19 '15 at 18:13