What's the probability of being perfect by accident?

So it's been almost 2 years since I said the random-number-machine follow-up video was coming "soon", and it's finally time! (Combined with a fun bit of math regarding perfect shuffles in reply to a fascinating Matt Parker video)

If you take a histogram of the wait times between geiger tube ticks, you get a beautiful exponential decay curve - most gaps between ticks are extremely fast, and it's rare to go a long time without a tick. You can use this property of random numbers to show that ticks coming from a tube are indeed randomly distributed.

Conveniently, runs of interleaved cards in shuffled decks of cards ALSO follow this exponential decay distribution, with short runs representing short wait times between ticks and very long runs of perfectly interleaved cards being rare, just like long wait times between geiger tube ticks. Because these properties of random numbers are so predictable, you can actually calculate the probability of executing a "perfect" or "Farro" shuffle by accident.

Music Credits, etc.: I Dunno by grapes is licensed under a Creative Commons Attribution license (https://creativecommons.org/licenses/by/3.0/) http://ccmixter.org/files/grapes/16626

Arcadia - Wonders by Kevin MacLeod is licensed under a Creative Commons Attribution license (https://creativecommons.org/licenses/by/4.0/) Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100326 Artist: http://incompetech.com/ ... https://www.youtube.com/watch?v=QzI6GFzH7ZA