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Suppose someone is asked to provide a long string of random digits, for example:

(just typed on keyboard, not generated in any way)

How good is it expected to be if used as random source?

For example, can I ask someone to think up a list of {1,2,3,4,5,6} numbers and use it instead of a dice (and expect each number to come with p=1/6 knowing the previous numbers).

Is it safe enough to just think up UUIDs without looking for UUID generator tools?

Is there some simple (not requiring any devices or pencil&paper) algorithm to improve "thought up" random sequences?

Update: Trying little experiment as suggested in one comment:

$ xxd -r -p > test

$ paq8l -4 test 
    Creating archive test.paq8l with 1 file(s)...
    test 250 -> 270    
    250 -> 292
    $ tail -n +3 test.paq8l  | wc -c
$ dd if=/dev/random of=test2 bs=10 count=25 iflag=fullblock 2> /dev/null
    $ paq8l -4 test2 
Creating archive test2.paq8l with 1 file(s)...
test2 250 -> 274         
250 -> 297

Both files (manualy typed and /dev/random-generated) seem to be uncompressible.

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This isn't really a mathematical question. It's more of a psychological one. – Qiaochu Yuan Dec 3 '12 at 19:49
I thought measuring the randomness/entropy is a mathematical thing. – Vi0 Dec 3 '12 at 19:57
This seems a bit like asking “how good are people at estimating the areas of triangles?” There’s certainly a mathematical aspect to it, but the main question itself is about how good humans are at something, so it’s essentially a psychological question. – Peter LeFanu Lumsdaine Dec 3 '12 at 20:20
There are statistical test suites to test quality of PRNGs - for example TestU01 or DieHard tests. – NumberFour Dec 3 '12 at 20:34
I don’t remember the source, but I’ve seen it reported that human-generated strings typically have fewer repetitions than would be expected of a randomly generated string. – Brian M. Scott Dec 3 '12 at 21:16

I think I cannot answer your first question (and I would say that this is generally hard to tell - depends whether the person really wants to generate random numbers - I wouldn't use this in some critical applications).

However - speaking of pencil&paper PRNG there is one developed by Bruce Schneier in his Solitaire encryption algorithm.

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The person wants to generate random thing (i.e. not trying to cheat anyway) yet does not calculate anything in head, just say numbers. – Vi0 Dec 3 '12 at 19:55
Well its too subjective - Myself, I think I cannot filter outter things while thinking numbers from the top of my head. You could try to think off some numbers, convert them to bytes and then try to compress it and compare it's size with the uncompressed version to get a rough entropy estimate. Truly random stuff is almost incompressible. – NumberFour Dec 3 '12 at 20:14
Tried to compress the file I typed. It does not seem to be compressible. – Vi0 Dec 4 '12 at 12:16

If a human be asked to select random number from 1 to 10, the number 7 would be selected with probability near to 1/3. So, don't believe human-generated random numbers. :-)

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Citation needed? – Pragabhava Dec 3 '12 at 20:59
Citation needed! – Did Dec 3 '12 at 21:15
This is experimental data from a Russian forum, hundreds of laymans participated in that poll. Unfortunately, I did not save the link to that thread. But you can easily reproduce this result by making the same poll in any forum not concerning with math: question = "select random number from 1 to 10", answer choices = 1, 2, .., 10. – Egor Skriptunoff Dec 3 '12 at 22:28

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