Probability: Drawing numbered lottery tickets

Question

There's a lottery with $1,2,3,4 ... n$ lottery tickets into a box and those lottery tickets are numbered from $1$ to $n$. We draw a lottery ticket randomly, and we note down its number and then we put this lottery ticket back into the box. We repeat this procedure $ k \ge 3 $ times. What's the probability of: a) Drawing the lottery ticket with the number "1" at least one time, and b) Drawing the "1", "2", "3" and "4" lottery tickets exactly one time each. Generally, i know there are ${n \choose k} $ ways to draw a ticket but except this, i don't have many other good ideas. Any help is apreciated.

Answer 1

a) $1-\left(1-\frac{1}{n}\right)^k$.

b) $4!\binom{k}{4}\left(\frac{1}{n}\right)^4\left(1-\frac{4}{n}\right)^{k-4}$.

Answer 2

You're sampling with replacement, hence each sample is independent from all others. For a fixed ticket, say "1", the probability of drawing it at least once is one minus the probability of not getting it on each try: $1-P(\mbox{not } 1)^k=1-\left(\frac{n-1}{n}\right)^k$