Your math gets you within an order of magnitude or so, check out NASA's work on ORDEM to predict this with Monte Carlos.

Two things: 1) The probability is very sensitive to # of objects and size assumptions. If you decrease the area of the bullets by an order of magnitude, the probability that it you will hit one 10 m^2 satellite in a year goes to 10%+ 2) satellite orbits are not randomly distributed. Within 700-900 km altitude, the probabilities are ~2-3 orders of magnitude higher (1/1000/m^2/y)

https://ntrs.nasa.gov/api/citations/20150003495/downloads/20...