As the title, I want to ask for some examples showing necessity and advantage of emperical definition of probability in teaching high school students.

We knew that the classical definition has some disadvantages, i.e, the sample space has to be finite and all cases are equally possible. Those are motivations for mathematicians to come up with the emperical definition. I would like to know some simple examples (to teach high school students) which we cannot apply the classical definition to calculate the probability.

Thank you very much.

  • $\begingroup$ It seems odd to me that people feel that probability should be a defined term. There doesn't seem to be any problem that needs to be solved and that is solved by these definitions. $\endgroup$ – Ben Crowell May 6 '17 at 22:53
  • $\begingroup$ What do you mean by the "empirical definition of probability"? Something like this? If so I'm not sure I understand the question, since empirical probabilities are the results of experiments, and can (basically) never be calculated theoretically. $\endgroup$ – user797 May 8 '17 at 22:20
  • $\begingroup$ If you're just looking for something where the equiprobable model doesn't work, consider flipping a biased coin. (I'd give this as an answer, but I'm not sure it actually answers what you want to know) $\endgroup$ – user797 May 8 '17 at 22:21

Consider any complicated real-world event. How likely is it that a given airplane crashes? Will Russia invade more parts of its neighbouring countries within 10 years? How far will an ant walk in a minute?

The classical definition utterly breaks in all of them. The third can be treated easily by repeated experiments (even as classroom activity, if one is so inclined, though maybe with a shorter time period). The second may not. The first one is a borderline case; should you use all airplanes or only the airplanes that share particular qualities?

I presume the ant example, or some other where the space of outcomes is continuous but easily observed, is what you are looking for. The others are more advanced and overshoot your goals.


Sports statistics should give plenty of examples.

For instance, what is the probability of a successful free throw in college basketball?

Here is a newspaper article discussing how this number has stayed remarkably consistent over the years: For Free Throws, 50 Years of Practice Is No Help.


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