[kccqzy]

Here's an even simpler example: a flip of a fair coin.

Suppose this is your first flip, one would intuitively think that there is 50% chance of H and 50% T.

Suppose you flipped once and got H. For your second flip, one might intuitively think that since the number of times getting H over the long run is 50% of the total flips, and we already have a flip of H, to "balance it out" the next flip should have a smaller probability of getting H. Instead that is wrong. The next flip still has 50% chance of H.

Suppose further that one has performed N flips, all of them H. One might even think that because of the way the geometric distribution works, it is very unlikely for the next flip to be H again. Instead that is wrong. The next flip still has 50% chance of H.