Zero probability

In a previous post I mentioned that an event can happen once or several times, although the probability of its happening is zero. The probability of an event is defined as the ratio of the number of favorable cases to that of possible cases. Therefore, if the number of possible cases is infinite, while that of favorable cases is finite, the probability turns out to be zero.

At first glance it seems incredible that an event with zero probability can actually happen. I think the matter will be clearer with a simple example. Two friends, A and B, are talking, and what they say is this:

A: If I ask you to choose a number between 1 and 100, what is the probability that you choose a specific number, such as 25?

B: 1/100, obviously.

A: If I ask you to choose a number between 1 and 1000, what is the probability that you choose 25?

B: 1/1000.

A: If I ask you to choose a number between 1 and 10,000, what is the probability that you choose 25?

B: 1/10,000.

A: If I ask you to choose a positive integer number, what is the probability that you choose 25?

B: Zero, for the set of integers has infinite elements, and one divided by infinity is equal to zero.

A: Choose any number among all the positive integers and tell me which number you have chosen.

B: I choose 2²⁵⁰⁰-1.

A: You have just made an event with zero  probability.

Thinking a little you’ll see that the probability of choosing, among all the integers, any finite set, however large, is also zero. For instance:

A: If I ask you to choose ten different numbers between one and one hundred, what is the probability that you choose precisely the numbers between 11 and 20? (their order does not matter)

B: 1 / 17,310,309,456,440

A: And if I ask you to choose ten different numbers among all the positive integers, what is the probability that you choose precisely the numbers between 11 and 20?

B: Zero.

I leave to the curious reader to compute why the probability of choosing numbers 11 to 20 among those from one to one hundred is precisely what B has stated.

To finish this post, I’ll propose a few more exercises for the reader. Whoever solves them has the opportunity to write a comment explaining how they arrived to the solution.

1.      What is the last digit of 6²⁵⁰⁰?

2.      What is the penultimate digit of 6²⁵⁰⁰?

3.      What is the penultimate digit of 6^1,000,000?

4.      What is the probability that the last digit of 6ⁿ is odd?

5.      What is the probability that the penultimate digit of 6ⁿ is odd?

By Vincent Pantaloni, CC BY-SA 4.0, Wikimedia Commons

The same post in Spanish
Thematic Thread on Statistics: Previous Next

Manuel Alfonseca

Happy summer holidays. See you by mid-August

Mathematical theology

Ernst Zermelo

Ernst Zermelo (1871-1953) was a famous mathematician of the early twentieth century. Among his achievements, the following can be mentioned:

• In 1899 he discovered Russell’s paradox, two years before Russell. Although he did not publish it, he did comment it with his colleagues at the University of Göttingen, such as David Hilbert. Russell’s paradox proved that Cantor’s set theory is inconsistent, since it makes it possible to build the set of all sets that don’t belong to themselves. There are sets that don’t belong to themselves, such as the set of even numbers, which is not an even number. Others do belong to themselves, such as the set of infinite sets, which is an infinite set. Now we can ask ourselves: Does the set of all the sets that don’t belong to themselves belong to itself? This question leads us to a paradox: if it does belong, it must belong; and if it doesn’t belong, it mustn’t belong.
• In 1904 he proved the well-ordering theorem as the first step towards proving the continuum hypothesis, the first of Hilbert’s 23 unsolved problems. The well-ordering theorem states that every set can be well-ordered, which means that any non-empty ordered subset must have a minimum element. To prove it, he proposed the axiom of choice, which we will discuss later.
• In 1905 he began to work on an axiomatic set theory. His system, improved in 1922 by Adolf Fraenkel, is a set of 8 axioms, which today is called the Zermelo-Fraenkel (ZF) system. Adding the axiom of choice to this system, we obtain the ZFC system, which is most used today in set theory.