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huh? How so?First of all, what you offer is not really a scientific hypothesis but just gambling. If one uses the word “hypothesis” loosely, it can be called one, sort of.
Why does this not have to do with confirmation (verification)? What do you think confirmation even means in a scientific context? I explicitly said in post #100 that,Now if you wish to establish your hypothesis, one experiment of tossing a coin a few times is not sufficient. What you say about the 50% chance of each coin toss, is of course accurate, but it has nothing to do with verification.
“Confirmation is raising the probability that the next **unexamined **case will be like the set of all **past examined **cases. And this task confirmation alone cannot do without an implicit assumption about natural laws from which predictions about the likelihood of future events within test scenarious can be deduced.”
So it simply doesn’t matter what the past probability distributions are. Flipping the coin an indefinite number of times never raises the likelihood the next coin will turn up head no matter what the probability distribution function actually is with respect to your past empirical frequencies in your “complicated statistical sampling.” But this is exactly what confirmation must do to make any sense, otherwise it’s just fluff in the wind. See next…
Yes, I am familiar with classical probability here. Thanks…What you need to learn about is statistical sampling, the different distributions stemming from the method of sample taking, chi-square testing, and whole lot of other, not too simple methods.
Here is the proper form of making a hypothesis of the coin toss. You examine the coin, see no obvious fault with it, and therefore hypothesize that it is a “good” coin. You wish to verify that assumption by making “n” coin tosses, and observe the distribution of heads and tails in the sequence. The coin tosses are independent events, in other words, the result of a toss does not influence the result of the next one (the opposite of this assumption is correctly called the Gambler’s Fallacy). We start with the fact that the probability of a head toss is “p” and the probability of a tail toss is “(1-p)”.
The null hypothesis is that “p” = 1/2.
No. Scientists don’t “deduce” the hypothesis from the evidence. The hypothesis is INDUCED from the statistical sampling and then subjected to further testing. This is precisely the issue at stake for which I mentioned concerning the problem of Induction in post #100 if you haven’t already read that. Under no circumstance does any number of past case trials raise the likelihood that the next case will conform to the frequencies of past cases without a prior assumption that regularities exist in Nature. So the “veracity” or likelihood of the general hypothesis that “the coin is a fair coin” is never raised over 0.5 with respect to the next unexamined case. You need this additional assumption in order to “raise the probability” from one instance to the next as you increase your number of trials in experiments.To establish the veracity of this hypothesis we decide to make “n” number of tosses. If our hypothesis is true, we shall see approximately “n/2” of heads, and “n/2” of tails. How do we -]deduct/-] [deduce] from this result that the coin is really “good”, or not?
Sure, this just deals with the converging of initial and posterior probabilities very similar to Bayes Theorem. But neither of them deals with the problem of induction.We set up a confidence level. In practice we usually use a 95% confidence, sometimes we might go up to 97.5% confidence. Now we look up a chi-square table, and find what kind of discrepancy we can expect from the “ideal” 1/2 value, and be still 95% sure that the coin was really “good”. This discrepancy is dependent on the number of tosses, obviously.
Precision is only a feature of probability densities, not of confirmation which is the raising of the probability that the next trial will be like your past trials.The larger the number of experiments is the more precise our verification process turns out to be.
It’s pretty much a red-herring, but I don’t think you meant that.Have fun. It might take a few years to learn the beuties of probability theory, but it is a good investment.
