Computer science randomized algorithms and probability with analysis

DUE MONDAY FEBRUARY 20th by 18:00. Please give detailed original answers(NOT COPIED FROM CHEGG) as to how you derived and proved each question. MUST give in-depth explanations for each question in words and mathematical notation showing how you prove each algorithms.In addition, you must NOT COPY THE ANSWER FROM CHEGG. I have had 2 people already try that.The same questions are on CHEGG website but the answers are wrong and IDK why people think it is ok to just copy the answers. Please take a look at the following pdf which are the notes from my class showing how the professor methodology on how he proves his algorithm There are 5 questions. Please look at the homework.pdf