what are the odds the middle flop card is the same 4 flops in a row?

Well, there are several ways to think about how "rare" something like this is.

(1) Consider seeing a flop. Middle card is 9d say. What is the probability that the middle card of the next three flops are all the 9d?

Let's leave aside any "card removal" effects of what cards more likely appear on flops (due to what hands players typically enter pots with preflop, there is a slight "bias" towards/against certain ranks appearing on flops).

Then the chance of getting the 9d for the middle card of the next three flops is simply (1/52)^3 = 1/140,608.

(2) Now broaden your horizon. Suppose you play 1,000 deals of a NLHE in the time period in question (say in a day or a week). Now ask what is the probability of observing four consecutive flops having the exact same middle card.

That answer is different than the first answer above, clearly. In 1,000 deals there are actually 997 sets of four consecutive flops possible (deals 1-4, 2-5, 3-6, ..., 997-1000). For simplicity, let's approximate this and call it 1,000.

In each of the sets of four consecutive flops, the probability of getting the exact same middle card is, as above, 1/140,608.

So you'd be tempted to now multiply (1/140,608) by 1,000 (our approximation of 997) to get 1,000/140,608 as your answer to the probability that there is some 4-sequence in there somewhere with the exact same middle flop card.

However, that multiplication only is valid if the underlying events are probabilistically independent. This is not the case in here. The chance that you get the exact same middle card in flops 23-26 is clearly not independent of the chance that you get the exact same middle card in flops 24-27.

To convince yourself of that, simply consider that flops 23-26 do have the exact same middle card (say the 9d). Now what would you say is the probability of flops 24-27 have the exact same middle card. Well, you know that flops 24-26 have the exact same middle card (9d), so all you need is for flop 27 to have that card (9d) in the middle.

Anyway, this is an interesting problem on which there has been a fair amount of work.

I encourage anyone who is interested in this problem or the math behind trying to solve it to post their thoughts.

Another angle is that if it were the left card instead of the middle card, you'd still be asking this question (but about the left card). So I think the more pertinent question is what's the probability of the same-positioned card being identical in 4 consecutive flops.

Good point well made.

For this problem (referred to above, not the OP problem which has been solved) I believe we are basically trying to figure out R.sub.p(r, n) discussed here where r=4, n=1000, and p=1/52.