  In a previous work, the first and third authors studied a random knot model
for all two-bridge knots using billiard table diagrams. Here we present a
closed formula for the distribution of the crossing numbers of such random
knots. We also show that the probability of any given knot appearing in this
model decays to zero at an exponential rate as the length of the billiard table
goes to infinity. This confirms a conjecture from the previous work.
