I think we calculate odds through 1/220 * 1/220 * 1/220

ChatGPT calculates odds at 1 in 10.65 million. Google just gives scientific notation.

For those exact hands in that exact order correct but realizing we would feel the same about JJ+ three times in a row overstates the odds of that happening by a great deal.

There are 2,652 possible ways to pick 2 cards from a standard 52-card deck. 6 of those ways are a pair of Jacks so your odds of getting a pair of Jacks would be 442 to 1. Getting Jacks 3 times in a row would be 86,350,888 to 1.

If you don't care about which pair you get, you have 78 ways to get a pair out of the 2,652 or 34 to one odds. Getting 3 pairs in a row is 39,304 to 1. If you already have a pair, the chances of you getting a pair in the next two hands is 1,156 to 1.

A pair is 1 in 17 not 34 to one. Easily calculated by you will get a first card all the time and 3 in the remaining 51 cards will make you a pair.

While statistically correct this is still a bit of flawed thinking.

If we instead calculate the probability of getting a sequence of this three hands: J2, 53, 84 we will find out that it's (16/1326)*(16/1326)*(16/1326) = 0,000175%. Or 1 in about 571 000 times. Indeed a higher probability, but still very improbable.

But nobody will ever say: "Wow I got J2, 53 and 84 in a row, that's just supposed to happen about 1 time out of half a million"

It's more useful to look at the probability of getting 3 pairs in a row like Polarbear1955 suggested. And that happens about 1 in 5 000 times. Often enough that online grinders will experience it on a regular basis.

DisRuptive1 made a very common mistake when it comes to probabilities: While it's true there are indeed 2 652 ways to pick 2 cards he forgot that each possible pair of jacks can be picked in two ways:

Jack of Diamonds, Jack of Hearts
Jack of Hearts, Jack of Diamonds

So in fact there are 12 ways out of 2 652. The reason we mostly say it's 6 out of 1326 is of course that it's of no relevance in NLHE which one of the two jacks the dealer gives you first.