Is these Negative binomial distributions?

#1
Hi.

Are these question the same(except the parameters of course) :

First - What's the number of trials needed until having at least 1 defective product, with a probability >= 85%, with pDef = 0.03.
Second - What's the number of trails needed until having at least 5 defective product, with a probability > 90% and pDef = 0.10

I understand the second is Negative binom, and its the first N that its Survival function of k=5 will result in over 90%.

Is the first one also Negative binomial and calculated the same? The reason I'm asking this is when I calculate the trail numbers needed(for the first one) like I explained above, I get 112, but if I use Geometric distribution, and check where the CDF equals to 0.85, I get 63. How is it possible that less trails are needed in order to have exactly 1 defective than the trails needed in order to have at least one. Doesn't "at least" include "exactly" ?

What am I missing?