When studying radioactive material, a nuclear engineer found that over $365$ days, $1,000,000$ radioactive atoms decayed to $976,596$ radioactive atoms, so $23,404$ atoms decayed during $365$ days.

a$.$ Find the mean number of radioactive atoms that decayed in a day.

b$.$ Find the probability that on a given day, $53$ radioactive atoms decayed.

a$.$ The mean number of radioactive atoms that decay per day is $($Round to three decimal places as needed.$)$

b$.$ The probability that on a given day, $53$ radioactive atoms decayed, is

$($Round to six decimal places as needed.$)$