Suppose that this is how a given casino’s 10-cent slot machine works: it has a random number generator which produces a string of numbers between 1 and 1000, given a seed value. Pulls of the lever are put into correspondence, chronologically, with this randomly-generated string. If a lever pull matches a certain designated number, say, 222, then that lever pull gets a payout of $90. Here’s a proposition about these slot machines:

A) The objective chance that the slot machine pays out, on any given pull, is 1/1000.

It’s true that, if we were to know the value of the seed and the nature of the random number generator, then we could figure out precisely when the machine will pay out. But, given determinism, precisely the same thing is true of any coin flip or die roll. Were we to know the precise microphysical initial conditions of the coin flip and the laws of nature, we could figure out whether the coin will land heads or tails. This is no obstacle to there being an objective chance associated with an event – it only tells us that a precise specification of the microphysical initial conditions is inadmissible information. Similarly, the seed-value and the method of random number generation is inadmissible information when it comes to the slot machine. But this doesn’t mean that there isn’t an objective chance that the slot machine pays out, on any given pull.

Here’s another proposition:

B) The objective chance that any given roll of a fair, six-faced die lands 1-up is 1/6.

This should be beyond reproach.

Finally, consider this proposition:

C) If there is a robust causal law to the effect that events of type A cause all and only events of type B — so that every A event leads to a B event, and no B event is caused by anything other than an A event — then the objective chance of an A event occurring is equal to the objective chance of a B event occurring.

Besides being intuitively plausible, I take (C) to be one of the central claims underlying the Bayes-Net approach to testing causal hypotheses. When we model causation and objective chance in the way specified by Pearl’s and Spirtes et. al.’s causal models, we allow the causal laws codified in the structural equations to induce a probability function over the endogenous variables. If (C) were false, then this would be illegitimate.

The Puzzle is that (A), (B), and (C) are inconsistent, as the following story demonstrates.

Suppose that the casino owners want to know the seed value for their slot machine. They want, that is, inadmissible information that will let them calculate, ahead of time, what their bottom line will look like after a certain number of pulls of the slot machine. However, while protective of their bottom line, they aren’t unscrupulous. They don’t want to plant the seed, they just want to know what it is. So, here’s what they do: they produce 6 randomly-selected seed values, using standard techniques (clipping three numbers from the end of a 10-digit decimal expansion of an arbitrarily selected time, e.g.). Then, they roll a die to determine which of these seed values will go into the slot machine.

Suppose that it’s true that, if the first seed is selected, then the slot machine will pay out on the 1001st pull of the lever. If any of the other seeds are selected, then the slot machine will not pay out on the 1001st pull. Then, there is a robust causal law asserting the following: The slot machine will pay out on the 1001st pull if and only if the die landed 1-up.

If (B) is true, then the objective chance of the die landing 1-up is 1/6. But then, if (C) is true, then the objective chance of the machine paying out on the 1001st pull is 1/6 — since there is a robust causal law saying that the die lands 1-up if and only if the 1001st pull pays out. By (C), the objective chance of the cause must be equal to the objective chance of the effect. So the objective chance of the machine paying out on the 1001st pull must be 1/6. But this contradicts (A), which says that the objective chance of the machine paying out on any given pull is 1/1000, not 1/6.

It’s true, of course, that both the die roll and the causal law involves all sorts of inadmissible information. But inadmissible information is only relevant to the question of what our credence should be. The puzzle, as I’ve formulated it, has absolutely nothing to do with credence. It has to do only with the objective chance function, and the connection between the objective chances of various events which are related by robust causal laws.