One game has just agreed and the flop for the next coming up. Whoa! You get a strange sense of deja vu. The flop is the same as the last flop. These things are supposed to be random. How can it happen twice in a row? Something must be up. But in fact such things happen so commonly that you should not even take note of them.
A mistaken calculation
What is the chance to get a particular flop twice in a row? First just ignore the card ordering and consider all the possible flop combinations. The order of the cards does not interest us since all the cards are flopped at once. There are 22100 combinations of 3 cards from a deck of 52. The chance to get one particular combination is thus 1 in 22100. So that chance to get it twice in a row is 22100 * 21100, or about 1 in 500 million.
Even given the volume of online play that is a rare event. Except that an error has been made in the calculation. An honest mistake that many people may not immediately see.
The solved question has been restated as "the chance to get a particular flop twice in a row" which is actually quite different from "a flop being the same as the one before". There is no particular starting flop in the desired calculation; Any flop will do. So to duplicate any particular flop has merely a 1 in 22100 chance.
That still seems rare
While a 1 in 22100 chance seems indefinitely still, it is by no means in the category of rare events. In the brick and mortar world hands are not dealt so fast, but in an online world a player can easily push through 50 hands even on a slow table.
A modest hobby player who puts in only 10 hours a week will end up seeing some 2000 hands in a month. After a year they will have put in over 24000 hands. We can reasonably expect that such a player will have seen this duplicate flop occurrence at least once in a year. Consider now a slightly more dedicated player putting in 2 hours a day and playing 4 tables simultaneously. At the lower rate of 50 hands per hand per table they will be getting in well over 100k hands per year. They can expect to see a duplicate flop about 5 times in the year.
It actually becomes even more likely for a multi-table player. They would reasonably consider any duplicate flops on any of their tables to be a match. That means for a four table player, on any flop there are not just 1 in 22100, but a 3 in 22100 chance that another table matches. Add to this that the player may consider the following flop on any of those tables as well, which now means a 7 in 22100 chance. So in the year this multi-table player can expect to see the duplicate flop about 35 times, or almost once a week.
Far too common to even notice
Despite the numbers an individual player may do a double take and make a comment about their duplicate flop. It does not mean anyone else will take notice however. As shown above even the modest player will see it often enough not to care. Consider further that popular online poker rooms can easily reach 20000 players. With this volume of players, even if they are all only playing a single table, the duplicate flop will happen to somebody about 50 times an hour, so almost once every minute.
At these volumes of play even the impossible becomes common. Instead of flops which have the same cards, what about the flops where the order of the cards is identical as well. Two flops with the exact same cards in the same order will certainly get a comment from a player. With the order kept, there are still only 132k different possible flops. So again with those 20k players playing 50 hands an hour, this will happen about 8 times an hour.
An even more unnecessarily case is a player receiving the same pocket cards cards and flop two times in a row. This has only a 1 in 26 million chance of occurring. For an individual player this is really rare indeed. Though here again, with a million hands being played online every hour this rarity occurs about once a day.
The final lesson here is that seemingly rare events happen quite frequently given the number of active players. Such information can be used to quell one's fears that the system is cheating or has a serious defect. This is no to say there are not really rare events that do infrequently happen. To brag about those one should do a few quick calculations first, lest they celebrate the mundane.