So, I'm looking through some basic probability stuff ahead of the Qualifying Exam. I'm just dusting it off; not expecting any revelations. That's pretty much how it went, but I did come across a curiosity. The text has a table summarizing the counts for the sample space for the four ways items can be sampled. It looks like this (with nicer formatting):
| Without Replacement | With Replacement | |
| Ordered | ||
| Unordered |
Perhaps it's just because I've been working with the Beta function a lot lately, but I had never noticed that when sampling with replacement without respect to order, the count can be expressed as r/B(r,n). It's not immediately obvious to me that there's any significance to that. After all, the Beta function is just a statement about combinatorics, so you could work it into a lot of counting formulas. But, step back from that and look at the formula and what it really means.
What we're saying is that if I was to randomly pick r balls out of a bag of n balls, replacing each time and you were betting on the sequence I'd produce, your odds of winning would be:
Is there any remotely intuitive reason to think that would be the answer?
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