IJK (iconjack) has calculated the probability of rolling a Yahtzee in no more than n rolls.
The first few numerical values are
p(1) = 0.0007716
p(2) = 0.0126316
p(3) = 0.0460286
The probability is 0.95 after 23 rolls, and 0.99 after 32 rolls.
Here’s a plot.
