Assuming you are drawing a number with replacement, each person choosing a number is a single independent experiment where the odds of success are 1/100, and the odds of failure are 99/100. So your distribution would be:
And you are trying to find P(X=3). It's really small. As it should be. For your second part, you are finding . This would be insane to do for x=3, 4, 5. . .100 - so simply calculate the compliment and subtract it from 1:
Here you are only solving three, so it shouldn't be that bad.