Lesson 15

How to interpret probabilities in real life? This is the object of study of mathematical statistics. For now, we will say just a few words about it. We assume that there is some experiment. The set of its potential outcomes is denoted by \(U\). We conduct the experiment many times and take \(\Pr[u]\) (for \(u \in U\)) as the proportion of all experiments that resulted in outcome \(u\).

If we keep in mind that we always imply the presence of an experiment (which is repeated many times) and its set of outcomes, it becomes easier to see statements and questions that do not relate to our mathematical model.

• “The probability of meeting a dinosaur on the street is \(1/2\): either we meet it, or we don't.” What is the experiment here? What is the sample space?


• “The probability that I have a one-hundred-ruble bill in my pocket is one-third.” What is the experiment here that we will repeat many times? We just need to check the pocket: either the bill is there or it isn't. If we want to say something about probability here, we should say that it is either zero or one.


• “Probably, he will be late.” This is hardly about probability theory in the mathematical sense.


• “It is quite probable that, in the 12th century, Arabian alchemists obtained the element using this process.” Again, what is the experiment here?


• “The number \((2^{5240707}-1)/75392810903\) is prime with a probability of at least \(0.99\). This is confirmed by a probabilistic primality test.” This number is either prime or not, there is no experiment here. The experiment in this case is the execution of the algorithm, which ended with the answer “prime”. We can estimate the probability that the algorithm gave the correct answer.