It seems like TV shows and movies about Mars are all the rage right now. In these films, the system designed to provide water is often breaking down, so let's build a model for such a setting. Suppose that the water system has a 1 in 100 chance of breaking down on a given day, and that breakdowns are independent from day to day. When a breakdown occurs, you fix it using a brilliant idea (as in the movies!) and then restart the system the next day. Sadly, your brilliant ideas are limited in number; you can only fix the system 4 times.

(a) Find the probability you'll have to use up all your brilliant fixes within the first 7 days of the journey. [WISE]

(b) How many days do you expect you'll go until you finally exhaust your last brilliant idea?

(c) Thinking in terms of the mean and standard deviation, how strange would it be for you to use your final fix on day 1400?

(d) Once the water system breaks down and you're out of ideas, you'll have enough water for 20 more days. How long should you expect to survive on Mars? [WISE]