Yes, that’s what I was thinking. A single riffle with no cut. That was the initial picture in my head. But you’re right that a “shuffle” could (and most likely) means the entire randomization process that includes a bunch of riffles and cuts. I’d actually never heard the word “riffle” before your post, to be honest. My word for that was also “shuffle” which may help explain the thinking behind my earlier post.
I guess another way to think about it is like lotto balls removed from a hopper. If you started with 52 balls and removed them one by one, the chance of the balls coming out in a previous pattern is not very likely.
A bingo hopper might be similar. It starts with 75 balls, but not all the balls are removed. Only enough balls are removed until someone wins bingo. But I would guess that there have been no repeats of bingo ball sequences in all the bingo games played.
Yeah, there are many ways to shuffle. The usual one you see is the “riffle” or “dovetail” where you separate the two decks into each hand and then interleave them. There’s also the overhand shuffle, which is also pretty common where one hand holds the cards, and the other hand grabs a bunch of those cards (usually 15-30 cards or so–they could be grabbed from the middle, they could be grabbed from the back, ideally you want to make sure you grab from both sections) and throws it over the top. A third way which is not seen professionally, is probably the first way one learned to shuffle, which is to throw all the cards in a big heap on the floor, mix them around, and gather them into a deck. This is usually only done with one pass, and I suspect does not produce the most randomized result.
Casinos do this at the blackjack table when they bring in new decks.
Shows you how often I gamble. 
And it’s called a ‘wash’/‘washing the deck’.
Handy hint: When you bring this up at your neighborhood poker game, and you’re getting really excited about it, making analogies for the hugeness of the number 52!.. do watch your audience. They may not care.
I cut my math lecture short when I noticed that no one was actually listening. (Happens a lot, I know the signs…)
BTW I was looking at this factorial table and I discovered a neat pattern. 5! equals 120 the first one ending in zero. Then 10! ends in 2 zeroes, 15! in 3 zeroes and so on. However at 25! you go from 4 zeroes to 6 zeroes. After that 30! ends on 7 zeroes, 35! in 8 zeroes and so on. Again at 50! you get a double-jump from 10 to 12 zeroes and again at 75!. I imagine this pattern continues and perhaps at 125! there is a triple jump and a quadruple jump at 625! ?
Is there a theorem which explores this pattern?
That’s not quite true. It was in a paper “Trailing the Dovetail Shuffle to its Lair” by Bayer and Diaconis. What they showed was that 7 shuffles puts the deck withing 1% of being truly random. They used, among other things, something called the shuffle idempotent that I was quite familiar with.
That must be what I’m thinking of.
Well, the number of zeroes is the number of times 10 goes into the factorial. So each power of 5 contributes that many zeros; there are at least that many 2’s.
I mean ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + … after counting them.
Winning lottery numbers have repeated often. Of course it’s only a sequence of a few numbers, not 52 of them. Also, the randomness of lottery drawings has been questioned. Those chosen with random number generators are most often suspected, I think mainly because the algorithms and actual implementations aren’t public while the ball selection can be watched live. However, actual cheating has been detected at least once with the ball machines.
On the other hand, a perfect Faro shuffle (a riffle where the deck is split in half and then each half perfectly interleaved card-by-card with the other half) isn’t very random at all, which is rather obvious if you think about it. In fact, 8 perfect Faro shuffles will return the deck to its original order.
Also, I think you would need to draw the repeated lottery numbers in the same order to complete the analogy.
True. So less likely to have happened, but doesn’t sound on the verge of impossible either, it is merely a few numbers though. I think that’s the point here, extending the possible permutations out to a sequence of 52 items, it is still astronomically redonkulous* to encounter the same sequence twice.
*redonkulous – “significantly more absurd than ridiculous, to an almost impossible degree.”
I don’t know if there’s a consistent formulation for these lotteries. A popular one where I’m from used to be based on 6 distinct numbers between 1 and 69 inclusive. For that lottery, there are 119,877,472 possible outcomes.
That may seem like a large number but it’s miniscule compared to 52!
That is interesting. It also demonstrates that evens with a probability of 1/52! are actually common.
Depends on how you mean.
Any particular arrangement of a deck of cards is 1/52! that you would get that particular arrangement.
But it is only interesting if you were trying to get that arrangement.
It’s the difference between getting a bullseye by hitting the center of the target and drawing the target around where your dart hit.
In this discussion I’m assuming “shuffle” means a number of ruffles intending to scramble the cards. I’m sure that there are magicians who can take a set of cards and shuffle them in an exact order every time.
It’s also interesting in that if low probability or even a mathematical probability of zero were to occur that declaring it impossible to be random is inaccurate.