Are 7 mash shuffles enough?

An extremely prevalent sentiment among most players is that mash-shuffling is not enough to evenly randomize the deck. They have the strong feeling that cards will not be "separated enough" from each other, and that if, for example (the most common example), they put all their lands on the top of the deck and start mash-shuffling, they will end up with huge clumps of lands, and huge clumps of non-lands (and thus they will either mana-flood or mana-screw). They feel that most certainly 7 mash shuffles (which is the oft-quoted figure) is not even nearly enough. For this reason they will often resort to pile-shuffling to "separate the cards". No matter how they are explained that 7 mash shuffles is indeed enough, and that nothing else is needed, they just can't accept it. They might accept something like 20 or 30 mashes, but certainly not 7.

But I'll try to explain it.

Let's assume that the deck uses decent sleeves (ie. not those cheap transparent thin sleeves intended for double-sleeving that can get sticky and pretty much glue cards together). Let's consider two cards that are adjacent in the deck, and let's examine how their relative separation will behave during mash shuffling.

You'd agree that with one mash, there's roughly speaking an approximately 50% chance that at least one new card will be inserted between them. Question: What are the chances that they will still be adjacent after 7 mash shuffles, on average?

Answer: Approximately 1 in 128, or about 0.8%.

Ok, there's only a very minuscule chance that they will still be adjacent after 7 mash shuffles. But they will still very likely to be very close to each other, right?

No. The thing is, the more separated they are already of each other, the more likely it is that more cards will be inserted between them. If you think about it, you'll realize that, in fact, their distance will, on average, roughly double with each mash shuffle. This means that their separation will grow exponentially with each mash.

What does this mean in practice? On the first mash, their separation will increase on average by 0.5 cards (which is what a 50% chance of a card being inserted between them means). This separation will, on average, double with each mash. Thus their average separation will roughly double the 0.5 seven times, ending up with an average separation of 64 cards. (And this is the average separation between them, not the maximum. The maximum could theoretically be anything.)

Since there is (typically) only 60 cards in the deck, a separation of over that means in practice that the card "wraps around" in the deck, to the other side. In practice this means that it will end up pretty much at a random place within the deck. There's a big chance that the relative order of the two original cards in the deck we were examining will switch (probably somewhere around the 50% figure, which is exactly what should happen.)

Yet, no matter how precisely this is explained, most players will not accept it. They will trust their own intuition more than mathematics. And they will waste everybody's time with useless pile-shuffling, and mash-shuffling dozens of times.