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Probability

Derren Brown recently had a show in which he flipped ten heads in a row. He just flipped coins all day and waited for it to happen eventually. If I flip a fair coin, I should believe there's a 50% chance it will come up heads. If I flip it three times, I should believe there's a 12.5% chance it will come up heads three times. If I have eight goes at flipping it three times, it seems I should believe there's a 100% chance of flipping three heads. If that's right, what's wrong with being increasingly confident at the beginning of each set of flips that this will be the one in which I flip three heads? It's obviously a bad argument: every time I fip the coin, there's a 50% chance it will turn up heads. But how could it be rational for me to bet that during the course of a day of coin flipping I'll flip three heads eventually but not be rational for me to be increasingly confident that the next set of three flips will be of three heads as the day progresses? Matthew

Yes, if I flip a fair coin 3 times I have a 1 in 2 3 (i.e. 1 in 8, i.e. 12.5%) chance of throwing three heads. How do we get that result? The rule is that if P and Q are independent events, then the chance of (P and Q) = chance of P x chance of Q. Likewise, if P, Q and R are independent events, then the chance of (P and Q and R) = chance of P x chance of Q x chance of R. If each of P, Q, R as a 1 in 2 chance, then the chance of (P and Q and R) is 1 in 2 3 . But, no, if I make 8 trials at throwing three heads I don't have a 100% chance of pulling it off. For the trials are independent events. And the chance of any one trial being successful is still 1 in 8, irrespective of what happened in the previous trials. Likewise, the chance of any one trial being un successful is 7 in 8, irrespective of what happened the previous trials. So the chance of eight trials being unsuccessful is (7/8) 8 , which is about 0.34. So the chance of getting three heads at least once in 8 trials is .66, i.e....
Probability

Does it make sense to talk of "probability" with regard to existential claims? Consider the following propositions: (1) Rolling snake eyes is improbable. (2) The existence of Big Foot is improbable. Though I can't quite finger the distinction, it seems to me that the notion of probability is being used very differently in (1) and (2).

Yes, different notions are indeed at stake here. We need to distinguish physical probabilities from evidential probabilities. Physical probabilities, also known as chances , are what are involved when we say, for example, that An atom of plutonium 238 has a 50/50 chance of decaying within 88 years. Smokers have a greater chance of getting lung cancer than non-smokers. The chance of rolling 1-1 with a particular throw of a pair of fair dice is 1/36. Note, the half-life of a plutonium atom is an objective physical property of it (a property it has independently of our beliefs about it). Likewise the probability of rolling "snake eyes" is a physical property of the chance set-up. And physical chance is related to another kind of physical property, namely the long-run frequency with which certain events turn up in a sufficient number of trials. For example, in the long-run, about 1 throw in 36 will turn up snake eyes. But philosophers argue over the relationship between the chance...