Well, how many verses have you collected and spent over the past 6 months?
It’s not all that bad if your overall bottom line looks something like this
1011 verse one
1025 verse two
1011 verse three
1021 verse four
That wouldn’t matter. The maximum number of GAPs I can do is equal to the lowest number of verses among the ones I have. So I can do 11. Then I would need to collect more of verse 1 and 3, but not 2 and 4. The fact that I have more than twice as many of two of them than two other ones is crazy and shouldn’t happen if they all have equal chances to drop.
That’s what you would think initially, but check this for the explanation:
Since there is no built-in corrective programming in GOW… probably never.
If only one verse was wildly different then it would easier (not easy, but easier) to correct as only a single verse would have to be found more often, or less often, depending on the case.
Once you have 2 verses wildly different from the average, the best you can hope for is that the verses you are short on show up on daily/arena offers.
Good luck with that 
My highest amount of verses is currently 66 higher than my lowest. Hope they’ll come up with something to fix this lol.
Ive got at least 30 each of 1, 2 and 3 and zero of verse 4. Ive had 1 GAP this week purely because i got a v4 offer from arena. As @igniteice says, ur only as rich as your lowest verse count but my verse count makes no sense whatsoever. Baz has been elusive since day 1 and i now pretty much ignore vault weekends cos without GAPs theres no point.
sls correct in the explanation.
you can run now 11 GaP and screenshot us 14 Verse 2, 10 Verse 4, and none in Verse 1, 3. and say that you never get Verse 1 and 3 or its multiple 10 times more in verse 2 and 4. but its not true.
even if a toss a dice 600 times, I won’t get 100 exactly of each side. so the probability is fine, but you must taking into account the overall GaP you already runned (like sls said) and check in chi-sqaure the probability chances. which should be fine.
When you approach one of those musician gnomes you have a 25% chance for each single gnome. That doesnt change even after thousands of gnomes. So you could end up with 10000 V1 and none of the others or it could be 2500 of each of those verses.
The probability is in both cases the same.
It’s a normal distribution, therefor it may happen, it may not. Probability is even on large data sets, but very clumpy on small scales.
Sorry, but this statement is very wrong… There is exactly one way to achieve 10000 verses of one kind and none of the others, but there are many, many, many (you possibly wouldn´t believe how many…) ways to achieve 2500 verses each. Obviously, this leads to a much higher probability (many, many, many times higher) for 2500 verses each.
I hope you are also aware of the “Law of large numbers” and its implications. Of course, it is a matter of definition and of the specific case, what a number “large enough” would mean. In our specific case of verses, where we have only four possible outcomes for every roll of the experiment, if a player has valid proof, that 10000 verse drops in a row were all one verse and none of the others, I would take that as a 100% proof, that the RNG of the game isn´t working correctly.
Please also take into consideration, that the law of big numbers is NOT saying, that after enough tries, the absolute frequency of the results will even out perfectly. It only says, that the relative frequency will align more and more to the expected values, when the number of repetitions is increased. In the very first reply to OP, @sls has already shown a perfect example, that shows this point.
Finally, the correct answer to OP in my opinion would be: Maybe the verse number will even out at one point (and after that deviate again), maybe not. However, what we can say for sure is: the more verses are collected, the lower the probability gets for having the exact same amount of all four verses.
10000 is a very large number when we factor in the rarity of verse gnomes to start with. so for most mortals, achieving exact parity is extremely unlikely. But to expect some degree of “equal distribution” appears reasonable until you consider how the gems RNG functions. if there is an equal chance for all 4 gnomes it doesn’t mean (like coin tossing) that you can expect reasonable parity any time soon. For that to happen the RNG would need to have “memory” so that it could “force” better equality by comparing verse count at various intervals. In my case with 30 plus of 1, 2 and 3 but zero v4 its fair to assume that this isnt addressed. Similarly to get a verse offer that completely ignores the existing imbalance is ludicrous because why spend gems on a verse which is already abundant in your collection?