For $9.95 a month, the subscription service MoviePass allows a subscriber to watch one movie per day at most movie theatres in the U.S. For each movie ticket used by a subscriber, MoviePass pays an average price of $9.00 to the movie theatres. MoviePass works at 91% of theatres in the U.S. and accounted for about 6% of the total domestic box office of $7.6 billion in 2018. MoviePass's business plan is to build a moviegoing "ecosystem" in which it profits from concession sales, rides to theatres, and advertising. Currently, MoviePass receives negligible income from such ancillary services. For the analysis of MoviePass's business model, assume there are 30 calendar days in each month and 360 calendar days each year. Data collected by MoviePass shows that, on each day, each subscriber has a 10% chance of watching one movie and a 90% chance of not watching any movies. Whether a subscriber watches movies or not on any given day is not affected by the watching patterns in the past.

1) What is the probability a subscriber watches exactly one movie over a month?

2) What is the expected monthly profit/loss from each subscriber?

3) What is the standard deviation of the monthly profit/loss from each subscriber?

4) What is the probability of earning a positive profit from a subscriber over a month?

5) Further analysis of MoviePass data shows that 20% of its subscribers are die-hard moviegoers, who have a 20% chance of watching a movie each day. The rest of its subscribers only watch a movie each day with a 7.5% chance. What is the probability that a die-hard moviegoer watches at least five movies in a month?

6) Due to its mounting cash deficit, MoviePass changed its service in August 2018. Under the new terms, subscribers can watch one movie per day and up to three movies per month. Subscribers buy movie tickets out of pocket if they exceed the quota. What is the expected profit/loss from a subscriber who is a die-hard moviegoer?

7) Under the new terms of service, what is the standard deviation of the monthly profit/loss from a subscriber who is a die-hard moviegoer?

8) Suppose a subscriber is not a die-hard moviegoer. Under the new terms of service, what is the probability that s/he needs to pay for movie tickets out of pocket at least once in the next year if s/he subscribes to MoviePass?

9) Under the new terms of service, what is the probability that a subscriber who is a die-hard moviegoer needs to pay for movie tickets out of pocket in at least ten months in the next year if s/he subscribes to MoviePass?