That's partially because MonkeyClub didn't do the math correctly. If the accident rate is really 1% of trips (which seems doubtful), there is a 99% chance of an accident free trip, a (99%)^N chance of N accident free trips, and a 1 - (99%)^N chance of one or more accidents occurring in N trips. For N = 100, this is 63%, and for N = 365*2 this is 99.9%.
As you mention, the data from the NHTSA seem to indicate a much lower accident/injury rate. Based on this data from the NHTSA [0], and going back to 2018 to avoid any pandemic-related effects on the statistics, it looks like there was on average only 1 (police-reported) crash per 481 thousand vehicle-miles, or one crash involving an injury or fatality per 1.7 million vehicle-miles.
[0] https://cdan.nhtsa.gov/tsftables/National%20Statistics.pdf, other formats and more data available here: https://cdan.nhtsa.gov/tsftables/tsfar.htm