At time of writing (Jan 2019), EmojiTetra chooses each new piece randomly with no consideration of previous pieces (there is no "bag" method). Thus there is a 1/7 chance of any given piece occurring next.
Thus, the chance of a run of, say, 5 I's is 1 in (1/7)^5, or 1 in 16807. The chance of a run of 8 I's would be 1 in 5.764801 million!
These both happened:
- Run of 8: https://twitter.com/EmojiTetra/status/1077177730540625920
- Run of 5: https://twitter.com/EmojiTetra/status/1058796612418363392
- See also: https://twitter.com/nemothorx/status/1080621003913351169
However, that is not the chance of that happening in general, only the chance of that specific piece happening at that time. If we don't care WHICH piece is involved in the run, then it's not about a run of "8 pieces which are I", but in fact a run of "7 pieces which are the same as the piece before it". That's 1 in 823542.
But what about the truly general question of "what is the chance of it happening at all, at any time, in a game of (say) length 1811 pieces (ie, game15, in which we had both the run of 5 and run of 8)
For that, it gets... complex...