In Baumann (American Philosophical Quarterly 42: 71–79, 2005) I argued that reflections on a variation of the Monty Hall problem throws a very general skeptical light on the idea of single-case probabilities. Levy (Synthese, forthcoming, 2007) puts forward some interesting objections which I answer here.

Bovens (2010) points out that there is a structural analogy between the Judy Benjamin problem (JB) and the Sleeping Beauty problem (SB). On grounds of this structural analogy, he argues that both should receive the same solution, viz. the posterior probability of the eastern region of the matrix in Table 1 should equal 1/3. Hence, P*(Red) = 1/3 in the JB and P*(Heads) = 1/3 in the SB. Bovens’s argument rests on a standard error in implementing Bayesian updating, which is (...) spelled out in Shafer 1985. When we are informed of some proposition, we do not only learn the proposition in question, but also that we have learned the proposition as one of the many propositions that we might have learned. The information is generated by a protocol, which determines the various propositions that we might learn. We should then update not on the proposition in question, but rather on the fact that we learned this proposition as one of the many propositions that we might have learned. A well-known application of this insight is the Monty Hall problem (MH) as Speed (1985: 276) points out in a discussion of Shafer 1985. As an illustration, let us apply Shafer’s insight to the MH. In the MH, the contestant in a game show learns that there is a goat behind two of three doors X, Y and Z and a car behind one door. She is asked to pick one of the three doors. The contestant picks door X. Monty will then open one of the remaining doors, which he knows to have a goat behind it. Suppose Monty opens door Y. The contestant is then asked whether she wants to …. (shrink)

In sum, then, Chalmers’s attempt to argue against physicalism based on the conceivability of zombies misses the mark. His version of conceivability does indeed imply possibility, but at the cost of making it unclear whether zombies are indeed conceivable.

Likelihoodists and Bayesians seem to have a fundamental disagreement about the proper probabilistic explication of relational (or contrastive) conceptions of evidential support (or confirmation). In this paper, I will survey some recent arguments and results in this area, with an eye toward pinpointing the nexus of the dispute. This will lead, first, to an important shift in the way the debate has been couched, and, second, to an alternative explication of relational support, which is in some sense a "middle way" (...) between Likelihoodism and Bayesianism. In the process, I will propose some new work for an old probability puzzle: the "Monty Hall" problem. (shrink)

This research examined choice behaviour and probability judgement in a counterintuitive reasoning problem called the Monty Hall problem (MHP). In Experiments 1 and 2 we examined whether learning from a simulated card game similar to the MHP affected how people solved the MHP. Results indicated that the experience with the card game affected participants' choice behaviour, in that participants selected to switch in the MHP. However, it did not affect their understanding of the objective probabilities. This suggests that there is (...) dissociation between implicit knowledge gained from the task and the explicit understanding as to why switching was the best strategy. In Experiment 3, the number of prizes and doors were manipulated to examine how participants construed the problem space of the MHP. Results revealed that participants partition the probability judgement to reflect the number of prizes over the number of unopened doors. (shrink)

Peter Baumann uses the Monty Hall game to demonstrate that probabilities cannot be meaningfully applied to individual games. Baumann draws from this first conclusion a second: in a single game, it is not necessarily rational to switch from the door that I have initially chosen to the door that Monty Hall did not open. After challenging Baumann’s particular arguments for these conclusions, I argue that there is a deeper problem with his position: it rests on the false assumption that what (...) justifies the switching strategy is its leading me to win a greater percentage of the time. In fact, what justifies the switching strategy is not any statistical result over the long run but rather the “causal structure” intrinsic to each individual game itself. Finally, I argue that an argument by Hilary Putnam will not help to save Baumann’s second conclusion above. (shrink)

The application of probabilistic arguments to rational decisions in a single case is a contentious philosophical issue which arises in various contexts. Some authors (e.g. Horgan, Philos Pap 24:209–222, 1995; Levy, Synthese 158:139–151, 2007) affirm the normative force of probabilistic arguments in single cases while others (Baumann, Am Philos Q 42:71–79, 2005; Synthese 162:265–273, 2008) deny it. I demonstrate that both sides do not give convincing arguments for their case and propose a new account of the relationship between probabilistic reasoning (...) and rational decisions. In particular, I elaborate a flaw in Baumann’s reductio of rational single-case decisions in a modified Monty Hall Problem. (shrink)