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Basically, I'm asking the chances of both sides getting Monotype teams. This happened to me in a Gen 7 Battle Factory match. I got all electric types, and my opponent got all dragons. Does this normally happen, or is it super rare?

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In Battle Factory, it is much more likely because all the Pokemon are from the same tier, but at the same time, it could also be impossible depending on the type.
It might be impossible to get a Monotype team in random battles because they're programmed to filter out "unbalanced" teams. You'll probably get a better answer if you ask here: https://www.smogon.com/forums/threads/ask-simple-questions-here-read-original-post-before-posting.3520646/
Give me a specific type then I can tell the chances of it.

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You never asked on Smogon, so I did it for you.

Gen 7 Battle Factory chooses randomly from 8 tiers including LC and Monotype, each with an equal chance of being chosen.

Gen 8 Random Battles allow 1 Pokemon of each type combination at most, and 2 sharing a type per team.
Monotype teams are impossible and tiers are very balanced with a certain Random chance of devalancing,

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Okay. Thanks.
There are now 2 people on Smogon arguing about this. Hopefully we can figure this out in the next few days.
Turns out it's currently impossible to get a Monotype team in random battles. I edited my answer.