Earlier today I posted a comment about searching through the decimal expansion of Pi for a given sequence. And, I gave an example of the sequence that the expansion contains.
Later in the day I thought:
What is the probability that the given length of the expansion of Pi contains a specified (albeit randomly chosen) sequence?
Consider the expansion of Pi to be a random sequence in that for a location the probability that the digit is d is 1/10. That's not really true, of course, but let's assume it anyway.
So, the first problem is to find the probability that the expansion of Pi contains any given random sequence. Let the length of the given sequence be variable. The length of the expansion is fixed.
The second part is to plot the probability of the expansion containing a random sequence of digits for an increasing length of the chosen sequence. Thus, the probability that the expansion contains a single digit (somewhere in the expansion) is 1. But what about the sequence "16" then, what about a sequence like "598" and continue on for longer sequences.
I don't have the answer, but I'm hoping you do.
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