I've been working on 2 problems which I really need help on: pretty sure that they are both binomial probability distribution problems. First one, I have the answer written down, but I think it is not correct. I don't really have clue on where to start with the second problem. Your help and advice will be greatly appreciated!
$1.$ You have $100$ coins, and $99$ of them are fair (equal probability of heads or tails). One of them is weighted and has a $90%$ probability of landing on heads. You randomly choose one of the $100$ coins. Find the probability that it is a weighted coin, under the following scenarios:
$1$) You flip it $10$ times and lands on heads $10$ times
$2$) You flip it $10$ times and it lands on heads $9$ times
A professor gives students a pop quiz with 5 true or false questions. Eighty percent of the students are well-prepared for the pop quiz, but twenty percent are not. Students who are prepared have a 90% chance of answering each question correctly, but the students who are unprepared simply randomly guess and have a 50% chance. Find the probability that a student was well-prepared under the following scenarios:
(a) Answered 1 correctly