Giving Your Knowledge Half a Chance

1000 fair causally isolated coins will be independently flipped tomorrow morning and you know this fact. I argue that the probability, conditional on your knowledge, that any coin will land tails is almost 1 if that coin in fact lands tails, and almost 0 if it in fact lands heads. I also show that the coin flips are not probabilistically independent given your knowledge. These results are uncomfortable for those, like Timothy Williamson, who take these probabilities to play a central role in their theorizing. Consider the following thought experiment. 1000 coins will be flipped tomorrow morning and you know this. You also know two further facts about these coins. Firstly, you know that the coins are fair – they have exactly as much objective chance of landing heads as tails. Secondly you know that they are causally isolated from one another – you know, let us suppose, that each of the coins are distributed over a number of different planets located in the Milky Way. There is no particular way of ordering the coins, but we may suppose that they have arbitrarily been assigned numbers between 1 and 1000 for convenience. Let Hn and Tn represent the propositions that the nth coin will land heads and tails respectively. The following question will be the focus of this paper: Question: what is the probability, given your knowledge, that a given coin will land tails. The notion of probability given your knowledge plays an important role in the theory developed by Williamson in [8] (chapter 10.) I will follow him in calling these ‘evidential probabilities’. The question, however, can be made intelligible without assuming the theoretical commitments of this particular theory. In order to make sense of this notion it suffices to assume that each agent is associated with at least one, and possibly a collection, of candidate ‘ur-priors’: probability functions that represent the agents opinions prior to obtaining any evidence at all. An agent’s evidential probability function at t is then determined by conditioning their prior probability function, Pr, on the conjunction ∗Many thanks to Kenny Easwaran, Jeremy Goodman and John Hawthorne for helpful discussions on earlier drafts of this paper. 1Indeed, some aspects of Williamson’s framework are completely unnecessary for this discussion. Most notably, his assumption that there is exactly one theoretically significant urprior.