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How rare is it to pick up a 10km egg?

There is the element of randomness, but it isn't all random. Like someone said, a single throw of the dice is random; but if you throw it over and over, the numbers will even out.

There are other factors, though. There is how many eggs you hatch, how many of pokestops you visit are ones that give 10 km eggs, and whether you have room when you pass those pokestops.

Since I've been using incubators, I find that with each round of eggs, I usually average 1 10km, 5-6 5km, and 2-3 2 km. But there is variation. Sometimes I will get 4 km and no 10km. On the other extreme, right now, I have 2 10km that are not far from hatching, and today I got another 10 km. Overall, though, the numbers average out.

The place I usually walk has 10 pokestops, and 3 give 10 km eggs, at least some of the time. I think that is about the average.
 
It seems like they've become a lot rarer, towards the start of the game there was a time where I had three in quick succession, but now I hardly get any. Niantic probably increased the rarity of them.

That is such a pain because of how common Eevee's are in 10K eggs.
 
I seem to be lucky with 10K eggs. Probably about 1 in 10 overall. No pattern at all lol.
That's around the rate I experience. I hatched my 60th through 63rd 10km eggs during the Halloween event. I've hatched 660 eggs total lifetime.
 
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