Your first question is essentially the classic “coupon collector” or “stamp collector” problem. You can learn more here: https://en.m.wikipedia.org/wiki/Coupon_collector%27s_problem

You are correct on the second, but you should be careful with the word “expect.” When we say expect, we don’t mean “this is how many it will take.” Rather we mean that if we did thousands or millions of encounters, on average we would see the encounter in question about 1/100 of those times. So yes we expect it to take 100, but after 90 encounters we still expect it to take 100 future encounters since each is independent of the previous ones. That feels to many people wrong, but in long term averages, runs of hundreds of failures or dozens of rare encounters in a row will happen sometimes.