Math puzzles!

Not 100% sure if this is proof but here it goes,
Given a random and infinite string of numbers N. Let's divide N into an infinite number of 9-digit numbers, let's call them Xi, where i from from 1 to infinite.
We now have an infinite number of 9-digit number strings that are all random. Since each string is random, and we have an infinite number of them we have a probability of 1 of at least one being 123456789. As a matter of fact we have an infinite number of 123456789 strings, but that's another story.

You're getting at it... but you have to explain WHY we have a probability of 1... not just "because there's an infinite amount of them"


Here's how (using your notation):

The probability of any Xi being 123456789 is 1/(1,000,000,000). So the probability that it isn't 123456789 is 999,999,999/1,000,000,000.

The probability that they ALL aren't Xi is:

limit x->inf (999,999,999/1,000,000,000)^x

Which, since the fraction is less than 1, is equal to zero.

Thus the probability that at least 1 of them is 123456789 is 1.


You pretty much had it.