Assuming the probablility of a royal is 1/40,000 the number of hands
corresponding to the first royal is (geometric distribution)
Percentile number of hands
50th 27,725.54065
75th 55,451.08129
90th 92,102.25242
95th 119,827.7931
99th 184,204.5048
There is a 1% chance that you may have to wait 184,204 hands to get
a royal. Further, the skewness of the distribution is apparent in
the difference between the expected meannumber of hands 40,000 and
the median number of hands (50th percentile) of 27,725.