Update

randomness.md

@@ -18,9 +18,9 @@ As with similarly wide terms, the word *randomness* and *random* may be defined
 
 Keep in mind **there are different "amounts" of randomness**, i.e. you should consider that **[probability distributions](probability_distribution.md)** exist and that some processes may be random only a little. It is not like there are only completely predictable and completely unpredictable systems, oftentimes we just have some small elements of chance or can at least estimate which outcomes are more likely. We see absolute randomness (i.e. complete unpredictability) only with uniform probability distribution, i.e. in variables in which all outcomes are equally likely -- for example rolling a dice. However in real life variables some values are usually more likely than others -- e.g. with adult human male height values such as 175 cm will be much more common than 200 cm; great many real life values actually have [normal distribution](normal_distribution.md) -- the one in which values around some center value are most common.
 
-**What do random numbers look like?** This is a tricky question. Let's now consider uniform probability distribution, i.e. "absolute randomness". When we see sequences of numbers such as `1, 2, 3, 4, 5, 6, 7`, `0, 0, 0, 0, 0, 0, 0, 0` or `9, 1, 4, 3, 9, 1, 5`, which are "random" and which not? Intuitively we would say the first two are not random because there is a clear pattern, while the third one looks pretty random. However consider that under our assumption of uniform probability distribution all of these sequences are equally likely to occur!  It is just that there are only very few sequences in which we recognize a common pattern compared to those that look to have no pattern, so we much more commonly see these sequences without a pattern coming out of random number generators and therefore we think the first two patterns are very unlikely to have come from a random source. Indeed they are, but the third, "random looking" sequence is equally unlikely (if you bet the numbers in lottery, you are still very unlikely to win), it just has great many weird looking siblings. You have to be careful, things around probability are great many times very unintuitive and tricky (see e.g. the famous [Monty Hall problem](monty_hall.md)).
+**What do random numbers look like?** This is a tricky question. Let's now consider uniform probability distribution, i.e. "absolute randomness". When we see sequences of numbers such as [1, 2, 3, 4, 5, 6, 7], [0, 0, 0, 0, 0, 0, 0, 0] or [9, 1, 4, 7, 8, 1, 5], which are "random" and which not? Intuitively we would say the first two are not random because there is a clear pattern, while the third one looks pretty random. However consider that under our assumption of uniform probability distribution all of these sequences are equally likely to occur!  It is just that there are only very few sequences in which we recognize a common pattern compared to those that look to have no pattern, so we much more commonly see these sequences without a pattern coming out of random number generators and therefore we think the first two patterns are very unlikely to have come from a random source. Indeed they are, but the third, "random looking" sequence is equally unlikely (if you bet the numbers in lottery, you are still very unlikely to win), it just has great many weird looking siblings. You have to be careful, things around probability are great many times very unintuitive and tricky (see e.g. the famous [Monty Hall problem](monty_hall.md)).
 
-Of course we cannot say just from the sequence alone if it was generated randomly or not, the sequences above may have been generated by true randomness or by pseudorandom generator -- we even see this is kind of stupid to ask. We should rather think about what we actually mean by asking whether the sequence is "random" -- to get meaningful answers we have to specify this first. If we formulate the question precisely, we may get precise answers. Sometimes we are looking for lack of patterns -- this can be tested by programs that look for patterns, e.g. [compression](compression.md) programs; number sequences that have regularities in them can be compressed well. We may examine the sequences [entropy](entropy.md) to say something about its "randomness". Mathematicians often like to ask "how likely is it that a sequence with these properties was generated by this model?", i.e. for example listening to signals from space and capturing some numeric sequence, we may compute its properties such as distribution of values in it and then we ask how likely is it that such sequence was generated by some natural source such exploding star or black hole? If we conclude this is very unlikely, we may say the signal was probably not generated randomly and may e.g. come from intelligent lifeforms.
+Of course we cannot say just from the sequence alone if it was generated randomly or not, the sequences above may have been generated by true randomness or by pseudorandom generator -- we even see this is sort of stupid to ask. We should rather think about what we actually mean by asking whether the sequence is "random" -- to get meaningful answers we have to specify this first. If we formulate the question precisely, we may get precise answers. Sometimes we are looking for lack of patterns -- this can be tested by programs that look for patterns, e.g. [compression](compression.md) programs; number sequences that have regularities in them can be compressed well. We may examine the sequences [entropy](entropy.md) to say something about its "randomness". Mathematicians often like to ask "how likely is it that a sequence with these properties was generated by this model?", i.e. for example listening to signals from space and capturing some numeric sequence, we may compute its properties such as distribution of values in it and then we ask how likely is it that such sequence was generated by some natural source such exploding star or black hole? If we conclude this is very unlikely, we may say the signal was probably not generated randomly and may e.g. come from intelligent lifeforms.
 
 TODO: moar