As there are infinitely many whole numbers, the experiment consisting of picking

Question

As there are infinitely many whole numbers, the experiment consisting of picking a whole number at random has zero modeled probability of picking any specific number. But if you actually perform the experiment and pick a number at random, the relative frequency of picking that number is not zero, even though there is a zero probability of picking it. Is this a paradox? Explain As there are infinitely many whole numbers, the experiment consisting of picking a whole number at random has zero modeled probability of picking any specific number. But if you actually perform the experiment and pick a number at random, the relative frequency of picking that number is not zero, even though there is a zero probability of picking it. Is this a paradox? Explain As there are infinitely many whole numbers, the experiment consisting of picking a whole number at random has zero modeled probability of picking any specific number. But if you actually perform the experiment and pick a number at random, the relative frequency of picking that number is not zero, even though there is a zero probability of picking it. Is this a paradox? Explain

Explanation / Answer

Here there are two misconceptions in the idea. First is that to pick something at random always implies that we pick something uniformly at random; and if we pick and object at random then that specific object will have a positive probability.

With regards to the first point, lets us consider that one cannot pick uniformly at random, say, a positive integer, since the probability Pk to pick K would not depend on K in N and the common value p of these probabilities Pk should solve the equation kNP=1, which has no solution.

To get around this paradox we make use of nonuniform distributions, like example a geometric distribution or a Poisson distribution. Then P k does depend on K and one can achieve the condition kNP=1.

Now coming to the second point, lets consider a uniform distribution in the interval [0,1][0,1]. And choose a random number U, such that xUy with probability yx for every 0x<y1. In particular, U1/2 happens with probability 1/2, 1/3U1/2happens with probability 1/21/3=1/6, and so on, but U=x happens with probability zero, for every x.

This type of observation led to the idea that additivity was a desirable feature of a probability, but only when restricted to countable collections.

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