This question is only marginally related to statistics, but is really a mathematical problem, a variation of a one-dimensional random walk.
If an event happens with a probability of exactly 10%, then the expected number of successes in 10 observation is exactly 1, so after 10, 20, 30 etc. observations it is possible that the number of successes matches the expected number of successes exactly. If the probability were for example exactly 10.03%, then an exact match would only be possible for the first time after 10,000 observations with exactly 1,003 successes. If the probability were π percent, then an exact match would be impossible.
If I understand you correctly: If you rolled a 10 sided dice for example 100 times, you would have ten opportunities that the number of successes matches the expected number exactly, after 10, 20, 30, ..., 100 dice rolls, and you are asking for the probability that this happened at least once.
Using the binomial distribution formula, you can calculate the probability that in ten rolls you have 0 successes (34.8678%), 1 success (38.7420%), 2 successes (19.3710%), 3 successes (5.7396%), 4 sucesses (1.1160%), 5 successes (0.1488%), 6 successes (0.0138%), 7 successes (0.0009%) or more (< 0.0001%).
Using the binomial distribution formula, you could also calculate quite easily the probability that after 10, 20, 30 ... rolls you have exactly the expected number of success, but that's not what you are asking for. You are asking for example for the probability that after 100 rolls, you have the exact right number of rolls at least once.
There's no simple formula for that. You can calculate that your chance is 38.7420% for an exact match after 10 rolls, 13.5085% for the first exact match after 20 rolls (either 0 success followed by 2 successes, or the other way round), 7.3269% for the first exact match after 30 rolls ((0,0,3) or (0,1,2) or (0,3,0) or (2,1,0) successes) and so on; a spreadsheet program will find the numbers that you want.
The name "random walk" comes from the fact that after every ten throws, the difference (expected number of success minus actual number of success) will randomly change. From memory, I believe in the one dimensional case you will always eventually get an exact match, but the expected number of throws for that is not limited. In the two dimensional case (that is if you had two dice and waited until they both simultaneously give an exact match) the probability that this happens eventually is less than once; again from memory the proof requires some rather hairy application of Fourier analysis.
The reason why it may take very long to get an exact match: Assume you rolled dice for some rather long time and you have 100 successes more than expected, which will happen sometimes. Now you continue rolling dice until the number of successes compared to the expected number changes by 100 again. Your chances are equal that the number is now exactly right, or that you have 200 successes more than expected. In the latter case it will take a lot more rolls to change that number either to 0 or 400, and so on.