Question

Show that it is not possible to choose a uniform positive integer at random. (In other words, we cannot define a probability measure on the positive integers that can be considered uniform). Hint: What would be the probability of choosing a particular number?

Answer 1

In probability theory, it is not possible to choose a uniform positive integer at random because the probability of choosing a particular number would be zero. This is due to there being infinitely many positive integers, making the probability of choosing any individual number infinitesimal.

Step-by-step explanation:

In probability theory, it is not possible to choose a uniform positive integer at random. The reason for this is that the probability of choosing a particular number from the set of positive integers would have to be zero, which contradicts the definition of a probability measure. In a uniform distribution, all outcomes have equal probabilities, but since there are infinitely many positive integers, the probability of choosing any individual number would be infinitesimal.

For example, let's say we want to define a probability measure on the positive integers. If the measure is uniform, then the probability of choosing the integer 1 would be 1 divided by infinity, which is 0. Similarly, the probability of choosing any other integer would also be 0. This means that the total probability would be the sum of infinitesimal probabilities, which would still be 0. Therefore, it is not possible to define a probability measure on the positive integers that can be considered uniform.

Answer 2

Answer: Think it this way:

If i told you to chose a positive integer at random, then all the positive integers have the same probability of being chosen.

this probability for each number is one (because you are choosing one number ) divided by the total of options in the set, and in this particular set, you have infinite options to chose.

so the probability of choosing one particular number is
`(1)/(infinity)  = 0`

if this is counterintuitive for you, think that numbers like 124125194104198172918274 are integers, when we think on integers the numbers that come to mind are 1, 2, 3.... etc, but there are a infinity of them.