Given an infinite series of truly random numbers, there are an infinite number of occurrences of streaks of n sequences. That is, an infinite number of monkeys with an infinite number of Microsoft Word installations will eventually reproduce the entire works of Shakespeare. The trick, though, is picking it out from the graft.
Well, to be precise, with probability 1 all that happens. But there is a distinction between having a probability of 1 and being guaranteed. (Think of tossing a sphere as rolling an “infinite-sided” die. Every particular point on it has probability 1 of not being landed on, yet in the end some point is picked all the same)
Though, as a technical note, one could, if one wanted to, consistently constrain one’s sample space to one where every “definable” property of probability 1 was guaranteed. (With the end result being that a tossed sphere, for example, will always end up landing on an “undefinable” point)
I hear that a million monkeys banging on a million keyboards will produce Microsoft standards complient IE releases on an average of once every five minutes. 
Anecdote:
When I was taking stats, my professor supplied a good number of us with TI-83 calculators. By coincidence (ha!), when we generated random numbers they were all the same. The reason is that the seed algorithm was the same on all the calculators.
With that in mind, are the numbers truly random, or is it a function of the program deriving the same numbers consistently, thus destroying the “randomness” of the computation?
OP, not sure if this is truly relevant in the eyes of the many mathematically talented Dopers we are fortunate enough to have; but I can tell you this:
(embarrassed but plucky, in the name of science) I like to play solitaire a lot on my pc. A lot. I have noticed this same sort of suspicious runs in the hands that are dealt, presumably at random. By way of illustration, I did 15 runs, just now. We are talking about the initial 7 cards which are dealt face up. Bear in mind that you are playing with only one deck of 52 cards.
In fifteen consecutive hands, four of these hands had seven different cards. However, of these four hands, two were consecutive, and they had four cards in common. (5,3,7,K,4,10,A // 3,5,7,4,Q,6,9)
Of the eleven hands remaining, every single one contained a pair. In addition, one of these hands had two pair; and two hands had a pair plus three-of-a-kind – in both cases, the triple cards were 5s.
It does this all the time. I have concluded (in a right-brained kind of way) that some randoms are more random than others. In other words, writing a program to generate random is difficult, the good ones are proprietary and hence expensive, & one probably gets randomness in direct proportion to how much one is willing to spend.
I could be totally wrong. It happens at least twice a year. 
True, but if you go in to test a (potentially) random sequence with that mindset, you’re going to trash every result you get since regardless of whether you run 1000 trials or 10 sets of 1000 trials, any answer you get is still entirely possible.
If I run a test of a coin-tosser and get 1000 heads in a row, you can damn well bet I’m going to be hesitant in declaring my coin-tosser capable of creating a random sequence. Yes, it’s theoretically possible for the event to have occured, but you’d be a fool to not run more tests.
The only time I use random numbers is for working out gambling things. Recently a guy at work told me about a betting system he planned to use on the races and I ised the random number generator to show him how huge he could expect some losing runs to be.
All I use is OpenOffice Calc and have a sheet set up with hundreds of columns worth of =RANDBETWEEN(1:100).
Each time you copy the formula into a new column you get a new list of numbers between 1 and 100 which can be ascribed results. Like for the guy at work 1 to 33 became winners as he expected that many. In the OP’s case 1 to 5 become kill missions.
Looking at the first 1543 rows of column 1 the longest sequences of numbers between 1 and 5 was 3, the next longest 2.
But to show how really random random numbers are the sequence of 3 was 4,4,4 and the sequence of 2 was 4,4.
Looking at the next 9 columns I got no sequences above 2 but one of kill, courier, kill, courier, kill, courier, kill, courier, kill.
All this talk of probability & RNGs seems grossly premature.
Just because the game (apparently) offers two kinds of missions, kill & courier, does NOT mean they have tbe distributed 50/50. The game designers could have chosen some other distribution on purpose.
Hell, I’d be willing to bet the distribution is grossly skewed. Something like 95% courier/5% kill missions, even. That seems like the kind of information that ought to be in the OP, though.
Isn’t 1500 missions actually a bit low to judge the randomness of the generator?
Did you read my post? I’ve said repeatedly that there is a basis for this hesitance, I’ve even outlined what it is (some kinds of “patterns” strike us as significant enough to allow inductive inference, while others don’t). My point has simply been that this basis is NOT the one of “Oh, that result is one which a random distribution assigns low probability”.
(If we were to get into the nitty-gritty of it, we probably model the coin-flipper as having an actual substantive IsRandom predicate, with our prior probability distribution on his flips being the independent draws from 50/50 one conditioned on the truth of IsRandom and being some particular very different distribution conditioned on the falsehood of IsRandom. Then, with the usual Bayes’ Theorem type wrangling, observations are capable of changing the probability we assign to IsRandom, the key being that P(observation | IsRandom) can differ from P(observation | ~IsRandom))
Lost the edit window: But whether or not P(observation | IsRandom) is itself low has little to do with it; all that matters is how P(observation | IsRandom) stands in comparison to P(observation | ~IsRandom) and P(IsRandom) itself (that is, to the probability that the observation would’ve occurred if the coin was not random, and to the prior probability that the coin was random in the first place)
Specifically, we’re looking for P(IsRandom | observation), which equals P(observation | IsRandom) * P(IsRandom)/[P(observation | IsRandom) * P(IsRandom) + P(observation | ~IsRandom) * (1 - P(IsRandom))]. You can see how this being low is an entirely different matter from P(observation | IsRandom) being low.
If only there were some way to find out…
They’re generated using an algorithm, so they aren’t random at all, but rather pseudo-random. There are tests that are run on random number generators that look at not just the numbers, but at sequential pairs, triples, etc. Some Intel chips (Pentiums of some sort, IIRC) had true random number generators in them (using thermal noise), but they were much much slower that psuedo-random number generators. I think the idea was they could be used in conjunction with PRNGs, but I’m not sure how much they were ever used, or if they are still in the chips.
To what extent is this apparent non-randomness caused simply by the fact that oddly streaky runs are more likely to be commented on than normally streaky runs?
-FrL-
Here, not much. The data are just too far of from what would be expected. The chance of getting 8 kill missions in a row out of 1543 missions is about 6 in 100,000,000. If it happened once, then yeah, your comment might possibly be a valid concern, but it happened twice.
One other possibility is that missions are generated according to some kind of hidden Markov model. It seems like kind of a silly way for the game designers to do things, but it’s possible, and if you’re really trying to figure out what’s going on, it has to be considered.
(I think it’s much more likely that someone tried to be clever and write their own RNG, but I thought I’d toss that out there.)
Since this is about Eve-Online, as a long-time player I have to add that the ratio (95/5) has been disputed for a long time. As far as I know, the devs have never actually stated the actual ratios for different divisions, whether they vary by agent level, faction, etc. That’s why you’ll often see conflicting ratios given by various sources.
For example, if the ratio was 70/30 at lower faction, and increased towards 99/1 at higher factions, you would expect to see long streaks like the OP, as completing missions would improve your faction, thus altering your ratio as you go. Overall, it might end up 95/5, but streaks of “kill missions” would be fairly commonplace early on.
The above is purely hypothetical, as there are many ways I can think of that could explain the streaks, while still maintaining a roughly 95/5 ratio overall.