When solving these problems remember the axioms of probability laws, particularly $0p(x)1.$ It's a good first$-$level check to perform. No problems require the use of a computer or calculator, however, wolframalpha can be helpful for visualizing some of the problems. For this assignment a computer or calculator may be needed to find certain values, which is allowed, however, it should be clear what expression is being computationally evaluated.

Bayes problem for refresher. Suppose you run a large $($but up$-$and$-$coming$)$ software company and provide a program that many people need. As is natural, sometimes things don't work right sometimes there's a programming error on your end, sometimes, however, it's truly user error. In this problem we'll apply Bayes' Theorem to figure out how likely it is something that we should change in the code.

For our program approximately $5\%$ of the code is not written properly and can cause malfunction $95\%$ of the time that part of the code is used the program crashes $($or worse$).$ From your user base, there's about $15\%$ of all your users complain that the program crashed. Assuming all portions of the code are used equally often, answer the following questions.

a$)(2$ points$)$ Since $15\%$ of all users complain of a crash, some of those are due to using the faulty code, while other crashes are due to user error. Find the probability of a crash, conditioned on none of the faulty code being used.

b$)(2$ points$)$ Sketch out the tree diagram describing the situation. The first branch is probability of using the faulty code, while the latter is probability of a crash.

c$)(2$ points$)$ Given that a crash occurred, what is the probability that it was due to software issues on your end?