Format change The regular and statistics podcasts are undergoing some changes. Stay tuned to this page for details. Today’s show is a little rushed because of this.
Randomness Why don’t we need to add “a fair die” or “an unbiased die” to our list of premises, “We toss a six-sided die, just one side will show, and just one side is labeled 6”? The probability of the conclusion “A 6 will show” is 1/6 without the addition of “unbiased”. If we add it, then our argument becomes circular, for “unbiased” means each side is equally likely.
We’ll talk more about the mysterious hold the word “random” has on experiments once we get to how to map our logical arguments to reality. That is, how do we determine whether a given die is biased or not?
Examples, finally! Give that three out of four dentists prefer Veldensteing Weissbier, what is the probability that your dentist prefers it? Obviously, 3/4. If you further know that your dentist is a teetotaler, then you must modify the entire argument to: “Given that three out of four dentists prefer Veldensteing Weissbier and my dentist is a teetotaler”, the probability that your dentist prefers that beer is 0—you have deduced it.
But you must always explicitly state your premises. You cannot criticize the conclusions of an argument for not including your favorite premises. Arguments stand on their own. Failure to heed this simple rule has led to more grief than anything else in understanding logic and probability. Research the sad story of Laplace’s Rule of Succession for an example.
Incidentally, in most of the arguments that matter to us (politics, religion, etc.), it is difficult to fully state our premises, a situation which makes it appear that probability is subjective. See the notes for more detail on this complicated subject.
Many multiculturalists are fond of saying, “There is no truth.” What is the probability the statement “There is no truth” is true? If the probability is 1, then the statement is false, for we have just found a truth. If the probability is 0, because there are no truths, then there are truths because the statement is false.
In other words, this is just another in a long list of asinine propositions put forward by half-wits who are anxious to get away with something.
But notice in our proof of the idiocy of that statement, we still used certain logical connectives, or steps, the validity of which we assumed true. We can never escape the fact that all truth eventually rests on our intuitions.
Last, my insurance company asked me to “prove that I do not have an additional policy with another company.” What is the probability I can prove this? Clearly state your premises.