The problem isn't that your math is wrong, so much as that you're misinterpreting it. The chance of getting the Vestments four weeks in a row is indeed (1/6)^4 = 0.08%, but that's also the chance of getting [i]any[/i] specific arrangement of four Warlock items over the same span. That is, there are 6^4 = 1,296 different permutations of items Xur could have offered, and one of them had to have been selected. This one sticks out not due to mathematics but rather due to cognitive bias. With that said, though, it's absolutely time for him to offer something else for us. As long as that something else isn't Sunbreakers.

Well technically yeah he could not be rng but it is possible. Yeah the math is a little off cause you are going off the odds of it happening for a total of 4 weeks. The system just sees the current week. So it will just be a simple 1/6 chance to get it each week. Just so happens that since it is completely and utterly random that it did the same one 4 weeks in a row. That is the point of having a random number generator. 1-6 isn't that big of a area of numbers to possibly generate. I'd understand if it was 1-100 or so.

Bungie confirms here that Xur's loot table is indeed RNG

Hopefully this will put this to bed (though not the question of whether Xur is random or not). There are 6 types of armor, as you said. Vestments, Praxic Fire, Nemesis, Veil, Ahamkaras Skull, Sunbreakers. You are correct that the odds that 6 straight Vestments is 1/6 * 1/6 * 1/6 * 1/6 * 1/6 * 1/6 So let's pretend it went PF, then Nemesis, then Skull, then Vestments, then Veil, then PF. What are the odds? The same 1/6 * 1/6 * 1/6 * 1/6 * 1/6 * 1/6. No matter what 6 events, the odds that those 6 occur in that order is 1/6 * 1/6 * 1/6 * 1/6 * 1/6 * 1/6. So your proof applies to anything he might've sold over 6 weeks. IOW, flipping a coin 3 times, it is equally unlikely it goes HHH as TTT as HTH or HHT, THT, or THH. Each of those are equally improbable. And one of those sequences has to happen. So whatever result you get flipping a coin three times, it was .5*.5*.5 to have the outcome over those three trials no matter what happened. This is why you cannot look at a random sample of 6 events. It's called variance. If something has a 10% chance of happening, it doesn't mean that in 10 attempts, it will happen once. It might happen every time ten times,, then not again over 500 attempts, but it will approach 10% the more trials. The difference between the results of the trials and the probability is shouldbe over infinite trials is called standard deviation. This gets people who play the lottery all the time. It is equally likely that 1111111 will get drawn as any other sequence of 6 numbers. But people don't think so. What you have proved is that any particular sequence of items for sale by Xur over 6 weeks is really unlikely.

I believe your math is good, but assumptions wrong. There are many different concepts in probability and I believe you are choosing the wrong event. The probability you described is the event in which times straight over a [i]specific[/i] time span. What I believe you should be looking at is the probability that the voidfang has been produced 4 weeks straight over a [i]general[/i] time span. Clearly as we go to infinity, the probability of this occurring at least once go to 1, regardless of it is random. In order to do this, the easiest way I can do it is to consider blocks of 4 weeks at a time, and let us assume that each week the item is chosen uniformly random over some set of size 6. Our first case to consider is the one in which the chosen block has all the same elements, this as you have said is 6^{-4}, very small indeed, helped only slightly by the fact there are 6 different possible ways to drop the same element 4 times, giving us a probability of 6^{-3}. But we must now consider drawing a block which matches with a previous block, what do these situations look like? Previous block ends with 3 identical the current block begins with that same element . 2 identical and 2 identical 1 element and 3 identical elements. That should be clear enough, so let us determine these probability that this chosen block will give us a string of four. We just need to add up the previous probabilities, which I will label with (x,y) where x will be the number of the 4 string on the first block and the y is the 4-x (0,4) - 6^{-3} as shown previously (1,3)- Because we do not want to encroach on (2,2) we must only consider solutions which are of the form xxBA there are 6^3*5 total sequences as such, giving us a probability of one of these occurring at 5/6. The probability that the currently chosen block matches it with 3 identical elements of that same type is 6^{-3}. Giving us a probability of 5* 6^{-4} (2,2)- Same thing, form must be xBAA, of which there are 6 * 5 * 6 = 6^2*5, giving us 5*6^{-2} probability of these occurring, and on the other side we have AAxx, which is 6^{-2}, giving us 5*6^{-4} (3,1)- BAAA- gives us 6*5 outcomes 5*6^{-3}, Axxx gives us 6^3 outcomes and 6^{-1} probability, so total is once again 5*6^{-4}. So the total probability of the currently placed block halting this is (6 + 3 * 5)*6^{-4} = 21*6^{-4} = something annoying. Anyway, because this is the probability that the current block chosen will halt the problem, we can now observe this like it is Bernoulli trial with independent events p except at n=1. So in conclusion the probability of this not occurring in n randomly chosen blocks is 1-(1-6^{-3}) * (1-21*6^{-4})^{n-1} . Because each block of n represents 4 weeks, and there has been about 16 weeks since the release of destiny, the actual probability of this having occurred over this time frame is around 1/20. Which by no stretch of the imagination is a large probability, but by no means unheard of. In conclusion, hunters rock.

Xur can't be random. Recently he's sold 3 guns which have never been sold; I believe it's a way of keeping people play the game, think about it raid resets on tuesday to keep you busy till friday then Xur comes. In my opinion exotics should never have been sold in the first place but what can you do? Nothing... (Not complaining, just saying)