Here are some additional goofy little stats facts about the cumulative probabilities: I went ahead and repeated this sequence of events (rolling 157 d20s) 10,000 times, and in all those cases, a Natural 20 count of 14 only happened 138 times. This means that the likelihood of rolling 14 Nat 20s with a perfectly balanced die is at best only 1.4%. In contrast, rolling 8 Nat 20s under those circumstances is about ten times more likely, occurring 14.4% of the time. Pretty unusual!
Rolling 10 Nat 1s out of 157 is a little less impressive, but still unlikely: the likelihood of such an event is 9.9%.
Thanks to @eponymous-rose for the additional statistics and explanation!