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Suppose I wanted to tell what's the probability of event $A$: getting 2 tails in a row of 5 coin tosses. According to the classical definition of probability, the probability of this event is equal to number of cases favorable to it divided by the number of all possible cases. What if I said there's only one favorable case (getting 2 tails), and there are 5 possible outcomes in total (getting 1 tail, 2 tails, 3 tails, 4 tails, 5 tails). The probability is equal to $\frac{1}{5}$.

How to prove a model is incorrect with respect to the experiment we want to model in terms of probability, like in the example above?

What is our certainty that the 'correct' answer is indeed correct? The 'right' answer here is $$P(A)=\frac{5\choose 2}{2^{5}}$$

I don't have any objection regarding this answer, but it doesn't mean it's correct, right? One person saying it's OK doesn't prove its correctness.

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Each event in the possible outcomes "1 tail, 2 tails, ..., 5 tails" is not equally possible (with the use of fair coin). So the "division by the number of possible cases" is meaningless in this situation.

In contrast, all events constitute the "right answer" is equally possible, and counting the number of such cases makes sense.

Of course, the standard (Kolmogorov's) probability theory is not the unique choice. The reason why we usually use it is that it accurately explains the physical phenomenon in our common world. It's just supported by experimental facts. In fact, in the nanoscale quantum mechanical world, this probability theory is not valid and we should adopt another system.

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  • $\begingroup$ You mentioned these outcomes are not equally possible. This is the source of major criticism of the classical definition of probability. What does equally possible mean? What tells you an event is equally possible? Intuition and common sense? Mathematics is not about intuition... I guess the right answer to my question is we cannot prove it, we must take the correctness of a model as an axiom. $\endgroup$ – user4205580 Apr 8 '15 at 13:53
  • $\begingroup$ I think you're right. We can't prove it. The only thing we can do is to make experiment in our physical world and verify the validity of our assumptions (equal possibility and a probability theory derived from it). In the pure mathematics, it's just an axiom as you said. $\endgroup$ – kohta Apr 8 '15 at 14:14
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In general, a “correct” law of probability is not unique when it is mathematically defined (a sigma-algebra on a set, a normalized mesure and other formalities; the ancient Bertrand paradox was a signal, “avant la lettre”, of this).

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