Throwing a die is an example of a physical act that corresponds so closely to a mathematical model that we can take a die throw as essentially being a mathematical event. Normally you would add in a rule to cover the case of something unusual or ambiguous happening, discarding that particular throw and generating another, so that a throw of a die is guaranteed to produce a result in a limited sample space. For example, rolling a six-sided die will result in one of the elements of the set {1,2,3,4,5,6}.Kevin Solway wrote:Firstly, I'd like to know exactly what it is about a dice throw that is "predictable" - and preferably also an example of how one can perfectly predict future events, instead of just guessing what they might be. Because if all you mean by "predictable" that we can guess what might happen in the future, then you aren't saying anything.
Throwing a die generates a sample from a finite group of integers (the "outcome space"), with a probability associated with each possible result such that the sum of the probabilities equals one. Based on the outcome of such samples, we might come to a conclusion about the die. For instance, if you roll a six-sided die ten times and come up with a six ten times, you can conclude that it is unlikely that the die is fair ("fair" meaning that there is an equal probability of each outcome in the sample space).
That you can not perfectly predict the outcome of a particular roll of the die is so obvious that is painful to mention it. But you can make statements about what you expect to happen, especially if you roll the die a large number of times. And that is often very useful.
I don't know why you are so down on "guessing." The whole point of all this is to be able to make very informed guesses, reflecting as much of the available information as possible. (And of course there are much more interesting cases than rolling a die.)
Next week: Bayes' theorem.