Peirce says funny things in a lot of places, though more often than not, what he says turns out to be interesting. It may be that he had too strong an empiricism/nominalism about probabilities at this stage in his thinking. And it may be that he was running together the two kinds of single-case problems.

I had the impression that the event was supposed to be problematic for statistical (frequentist) probabilities in the same way that an event such as China winning the race to land a man on Mars is, but this is evidently not the case. Card draws qualify as a “mass phenomenon” (to use von Mises term), allow, in principle, for unlimited repetition, we supposedly have actual statistical data supporting our assigning the probability 1/26 to the event of drawing the black card from the red deck, and so on.

But of course von Mises (I don’t know Peirce’s theory very well so use him as the prototypical frequentist) had another problem with assigning probabilities to particular events, even statistical events, in that it seemed to him that the probability could be anything depending on the choice of reference class. I suppose the event in Peirce’s thought experiment could be regarded as problematic in this sense. But it’s not perfectly obvious to me that it is. We are allowed, on the frequentist picture, to say that the probability of drawing some particular card from some shuffled deck of 26 cards is 1/26. To me this seems to go far in justifying our claim about what is the rational choice in the described situation.

Peirce says something funny in one place: “But in the case supposed, which has no parallel as far as this man is concerned…” This seems to suggest that only statistical data that has been personally and directly acquired qualifies. This isn’t what other frequentists had in mind at all. Frequentist probabilities are supposedly objective, and the underlying data can be shared. Bracketing the reference class problem, I am “frequentistically justified” in believing that the probability of drawing a particular card from a shuffled deck of 26 is 1/26, even if I personally only ever make one such draw.

Well, sorry it didn’t work out for you, but at least you acted in accord with reason. Unfortunately, the rational act was not linked with certainty to the more desirable outcome; the act was fully rational (in light of the respective probabilities and utilities) but that doesn’t mean that the desired outcome was fully certain. Now have fun in hell!

Again, not much of a consolation I suspect! But as soon as probabilities enter the picture, it seems that we must allow for the most rational act not always leading to the most desired outcome.

The question, then, is about why we say that picking from the mostly white deck is the rational thing to do, or why we say buying houses/hotels is the rational thing to do. And a big part of answering the challenge is figuring out what to say to someone who has done the rational thing and lost big.

Does that make sense or am I confused? Incidentally, I’d still like to know how people think the puzzle is to be resolved. What do you think, Jonah?

Peirce seems to be concerned about the psychological impact on the petitioner who despite his best efforts at applying reason and the presumed very high propensity of the single trial to go his way nevertheless loses the game and suffers an undeserved disastrous consequence. With appropriate tailoring, Peirce’s scenario is analogous to the recent financial manipulations involving high risk mortgages and associated derivative financial products. Except, of course, instead of the financial players (who gambled that a very low probability event wouldn’t occur in a single “trial” and lost) deservedly going to psychological and financial hell, (relatively) innocent bystanders are instead. Unlike the bankers, Peirce’s petitioner apparently isn’t well-connected with members of heaven’s admission committee who can pull a few strings on his behalf notwithstanding his bad luck.