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Old 2022-05-05, 23:39   #1
Graph2022
 
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Minus Does it sound random? - Regards Perfect numbers

Hi Guys,

New to forum so sorry in advance for "stupid questions" :)
Does below table look random? I checked 46 Mersenne primes, and below my some ,,tweaking".
Shouldn't it be evenly split assuming it is random?
The sample is small (46) but there are 32 options, so there should be 1 at least for each on average, but it is not the case.

Thanks

1 1
2 1
3 2
4 3
5 1
6 1
7 0
8 4
9 1
10 1
11 0
12 2
13 2
14 1
15 0
16 2
17 3
18 0
19 2
20 3
21 2
22 2
23 1
24 0
25 0
26 0
27 0
28 4
29 2
30 3
31 0
32 2
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Old 2022-05-05, 23:53   #2
slandrum
 
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I have no idea what you are talking about, or what the table you are showing represents.
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Old 2022-05-05, 23:53   #3
Batalov
 
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Quote:
Originally Posted by Graph2022 View Post
Shouldn't it be evenly split assuming it is random?
The sample is small (46) but there are 32 options
Should what exactly be evenly split assuming whatever?

More generally (regardless of what your answer will be):
Sampling patently random output into 32 bins and having only 46 inputs - what sort of "randomness" in bins would you even expect to find? Replace your inputs with totally random inputs. What will you observe?
Even before reading smart books on sampling/bootstrapping, you can cheaply learn the shape of a distribution by repeating the process N times (e.g. 10^6 times). It is very easy.
Hint: the result will be that you cannot reasonably expect totally flat distribution, forget about "at least one in each bin". That is if you draw 46 samples by some rule into 32 bins.
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Old 2022-05-06, 00:15   #4
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As an aside - this is a very common thing that happens in the beginner card players (even though I call them beginners, most of the people I know never actually leave this stage!) -
they frequently complain "I've got a bad hand (of cards)" or "Look, I've got a bad hand five (ten) times in a row!", and then invariably "The dealer is cheating!" or (because most game rooms are now virtual) "The random generator is broken!"

The truth is - they have no idea what random sampling should look like. They think that getting too many (in their opinion) of Jacks, or too many times in a row three Jacks at a time in consecutive draws, or whatever.

The baseline for anyone who wants to approach statistics is too observe random data for a definite amount of time. (In case of cards, if one doesn't have enough experience or partners, - to buy a deck and deal for a few hundred times every evening for a week straight. Only then they will /maybe/ stop saying "This hand is not random!" )
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Old 2022-05-06, 00:15   #5
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If you have a perfectly random method of picking a number from 1-32 with an equal chance of each choice, then after 46 independent picks any given number has about a 23% chance of not having been picked. In other words, you'd expect about 7-8 numbers to not have been selected after 46 trials, but it wouldn't be surprising if it were 6 or 9 that weren't picked. What would be extremely surprising is if all 32 numbers had been picked.

What this has to do with Mersenne primes or perfect numbers is not at all clear.

Last fiddled with by slandrum on 2022-05-06 at 00:19
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Old 2022-05-06, 00:21   #6
Graph2022
 
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Slandrum, any reference / topic on how 23% chance was calculated?

Thanks

Extra question not quite related to topic but to Mersenes. How long does it take to generate 2^470,000,000 in text?

Last fiddled with by Graph2022 on 2022-05-06 at 00:24
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Old 2022-05-06, 00:29   #7
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if a number has a 1/32 chance of being picked, then it has a 31/32 chance of not being picked on each trial. (31/32)^46 is about .23213
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Old 2022-05-06, 01:26   #8
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Quote:
Originally Posted by Graph2022 View Post
<snip>
The sample is small (46) but there are 32 options, so there should be 1 at least for each on average, but it is not the case.
<snip>
32 options for what, exactly?
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Old 2022-05-06, 01:29   #9
Graph2022
 
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Quote:
Originally Posted by Dr Sardonicus View Post
32 options for what, exactly?
Thread already answered for me by Mr. Slandrum.

Thank you!
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Old 2022-05-06, 02:01   #10
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Quote:
Originally Posted by Graph2022 View Post
Thread already answered for me by Mr. Slandrum.
Actually, the question hasn't even been asked yet. You haven't explained what the 32 options are.

You've apparently got some "random variable" which assumes the values 1 to 32. You seem to want to know - Is the distribution uniform? Your results with a sample size of 46 don't seem terribly unlikely assuming uniform distribution, but neither can they be considered confirmatory - the sample size is too small. It's sort of like asking if a coin is fair, based on the results of 3 tosses.

Knowing what your 32 options were - that is, what your random variable actually is - might give another way to test whether it "should" be uniform.
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Old 2022-05-06, 02:19   #11
LaurV
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I think that what OP tries to say, is that, if you have a random method to pick numbers, for which the output is one in 32 possibilities (like, taking random numbers mod 32, for example), you would expect after an "enough large" sampling process to get all 32 buckets populated in a reasonable equal amounts.

Of course, this relies on the numbers being really random. For the example with "mod 32" which I gave, if you pick prime exponents, obviously you will only get 16 buckets filled (the odd buckets, as there is no odd prime to give an even residue mod 32). Some more restrictive method may only fill fewer buckets, for all primes (think Euler test, as the extreme, it will fill all buckets for random numbers, but fill only first bucket for primes).

What the OP should do, as the other posters suggested too, is to take his method and shove it to his butt and apply it to all the prime exponents in some range, and see if all 32 buckets are filled in about equal quantities. Then apply it only to the exponents known to generate prime mersenne numbers (which I assume he already did in post #1 above), and if that indeed fills only the first 5 buckets, as the first post shows, then yeah, you are indeed onto something big ...

Last fiddled with by LaurV on 2022-05-06 at 02:25
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