Natural probability

Recently, I’ve come across a nice question: what is the probability that randomly chosen natural number ends with digit 5?

Let’s solve somewhat easier question, first: what is the probability that randomly chosen number between 1 and 10 ends with digit 5? There are 10 numbers in total while only number 5 satisfies the condition, therefore 1/10 = 0.1 is the right answer.

Unfortunately, getting back to the original question, there are infinitely many natural numbers ending with 5. And what’s worse, count of such numbers is the same as count of all natural numbers. That’s just a property of countable sets. Therefore we can’t apply our original idea to all natural numbers.

Or, can we?

Natural numbers have amazing tool called mathematical induction which shows to solve our problem pretty easily.

Let’s denote:

• m[1] = {1, 2, …, 10}
  probability of choosing number that ends with 5 in m[1] is 0.1, which is just our previous result
• m[2] = {1, 2, …, 20}
  probability of choosing number that ends with 5 in m[2] is 0.1, no surprise here
• m[i] = {1, 2, …, 10*i}
  probability of choosing number that ends with 5 in m[i] is 0.1

And since we can construct m[i+1] from m[i] by adding 10 following numbers where only one of them ends with 5, probability in m[i+1] is 0.1, as well. Mathematical induction is our friend.

Sequence of sets m[i] is clearly countable and property of 0.1 probability is induced over the sequence, hence over all natural numbers. The answer to the original question is 0.1

There’s a ton of very interesting topics that could be discussed further.

What’s beautiful about countable sets is the following result. Regardless of the starting number, 0 or 1 or 4 or 1000000, the probability will always remain 0.1. All we need is to construct m[1] properly, e.g. for 14 we get m[1] = {14, 15, …, 23}, m[2] = {14, …, 33}, … you see? It’s 0.1 again.

Since the size of events A={i; i ends with 5} is same as B={i; i is natural number}, usual ratio p=|A|/|B| would give us probability of 1.0, which is nonsense. Well, that happens if you are not careful about size of events, |A| and |B| are not natural ordinals and have a bit different algebra.

See you soon.