裁判員になる確率は1/100です - www.textfile.org

883840239413745673888484367614071057790536685391943120866342686853088554088962741529800523812714432924482598710613913144352575140063676571506082961578643064214433481064949969375249999 / 88817841970012523233890533447265625000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
で約 0.00995115643219699 ですかね。
Ruby 先生に解いてもらいました。

def f(rat, times)
  d = rat.denominator
  n = rat.numerator
  result = n
  (1...times).each do |i|
    result = n * d ** i + (d - n) * result
  end
  result.quo(d ** times)
end

puts f(1.quo(5000), 50)

短く。

n, d, times = 1, 5000, 50
puts times.times.inject(0) {|r, i| n * d ** i + (d - n) * r }.quo(d ** times)