Problem 1: What is the probability of rolling two standard $n$-sided dice such that the numbers on top of them sum to $k$?
Problem 2: Suppose you have exactly $n$ lottery tickets. Each lottery ticket can either be a winner or loser but not both. The probability that a lottery ticket is a winner is $p$ and is independent of the other lottery tickets. What is the probability that exactly $k$ of your lottery tickets are winners?
Problem 3: Suppose you have exactly $n$ lottery tickets. Each lottery ticket can either be a winner or loser but not both. The probability that a lottery ticket is a winner is $p$ and is independent of the other lottery tickets. What is the probability that at most $k$ of your lottery tickets are winners?
Problem 4: You are taking care of exactly $n$ dogs. You let them go play without their name collars. After the dogs return, you forgot their names, so you put the collars back on randomly. No two of your dogs have the same name. What is the probability that exactly $k$ of the dogs have their correct name collars?
Problem 5: Two people each flip a fair coin exactly $n$ times. What is the probability that their coins landed on heads the same number of times?
Problem 6: You flip a fair coin exactly as many times as it takes until the coin lands on tails. What is the probability that you flipped your coin $k$ times?
Problem 7: You flip a fair coin exactly $n$ times. What is the probability that your coin landed on heads exactly $k$ times?