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# probability of a repeated decimal

## Question

What is the probability that you could pick at random from the rational numbers a number that contains the string ....7777777..... (the string of six 7's need only to exist consecutively while other digits can occur in the number)?

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I'm thinking the answer is different for rational and real numbers.
Is your question purposely directed to rationals?

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15 hours ago, bonanova said:

I'm thinking the answer is different for rational and real numbers.
Is your question purposely directed to rationals?

It is purposefully directed but not meant to imply that there is a different result.

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If we consider only integers, from zero to infinity, and we could list them all, we could see that the quantity of low-value-numbers is negligible when compared to the quantity of very-large-valued numbers. Thus we can safely assume that when randomizing, we will pick only very-large-valued numbers.

It is also said that randomizing 0-9 and combining them to form enormously (infinitely) large number, every combination of strings (including ..7777777..) is contained in that number.

So, since we can safely assume that we will be able to pick only very-large-valued numbers, and in each of those numbers, the string 7777777 is contained, We can assume that every time we pick a number randomly, the string 7777777 is contained in that number.

Thus, 100%. Also the same for negative numbers. And it won't matter if they are integers or not, you can use the same idea for all rationals.

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