This a real world problem I am working on but I need some help. The candles are camouflage but the problem is real.
There are X candles that are broken in to two halves. X(max) is somewhere between 24 - 72, that is 48 and 144 halves. 48 halves have been found: (exactly 24 top halves and 24 bottom halves). It is not known for sure if any are missing.
Also, it is not known if any of the 48 found halves are a match for each other.
Here's the problem: X(max) should be 72 candles (144 halves) But we can only find 24 candles (48 halves) and we have found an equal quantity of top and bottom halves. We suspect that there was only ever 24 candles and not 72. How can I prove this mathematically? I can sense it to be a probability problem but I am at a loss.
What I mean by my original comment is this: different mathematical procedures lead to different mathematical results (i.e. different precise mathematical statements) but people have different preferences re: which result they consider "acceptable" as "proof" that there aren't 72 candles.
If Bayesian is too complicated, try https://en.wikipedia.org/wiki/Statistical_hypothesis_testing
The basic idea of hypothesis testing is that you first ASSUME there are indeed 72 candles (so called "null hypothesis"), then you calculate the probability that, assuming there were indeed 72 candles, yet you only find 48 halves. If this probability is very small, you "conclude" there can't be 72 candles after all. (I put "conclude" in quotes because many people would not agree this is a "conclusive proof".)
How you calculate the probability is up to you. But at minimum you need to a value $p$ for "probability that, despite looking and looking, I cant find a 49th end". This is domain specific as I have no idea how hard you looked.
Another assumption you can make is to assume each end (out of 144 ends) is equally like to be found next. So you can calculate the probability that the first 48 ends to be found happen to be 24 tops and 24 bottoms. This has a precise answer (under all the assumptions stated) $= {72 \choose 24}{72 \choose 24} / {144 \choose 48} \approx 0.14$. I.e. if the null hypothesis is true, your chance of finding 24 tops and 24 bottoms (among the first 48 ends to be found) is 14%.
If this number were much lower (e.g. 1%) some people may reject the null hypothesis already, by reasoning that they couldn't have been so (un)lucky as to hit a 1% chance event. However 14% is high enough that people usually won't reject the null hypothesis due to this alone. You would need help from the probability $p$ defined earlier.
So that's the general idea. You assume the null hypothesis (in this case, 72 candles), then if you can show that probability of observing what you observed is "very low" (pick your own threshold), you reject the null hypothesis by saying "hey I couldn't have been so (un)lucky as to hit that small a probability". As I said, people disagree as to the acceptability of this reasoning as "proof". Also, note that this does NOT let you conclude that therefore there are only 24 candles (unless you somehow know it must have been one of two cases only, and not e.g. 25 candles).
Hope this helps? Read the wiki.