Alex is writing an exam for his math class, and he decided that he was going to do a good problem relating to chapter 6 or chapter 7 (or both) to put on the exam. In particular, he has 20 problems to choose from, including 12 good problems involving chapter 6 material and 15 good problems involving chapter 7 material (some problems involve material from both chapters).

a) How many of the problems are related to chapter 6 but not chapter 7? Show your work please step by step. ( Start by finding the number of problems related to BOTH chapters and then piece it together from there!)

b) Suppose that Alex rolls a customized 20-sided fair die to decide on which problem to put on the exam. What is the probability that he will choose a problem which involves material from chapter 6 but not chapter 7? Show your work please step by step

c) Suppose that Alex instead rolls a different customized 20-sided die which he has rigged to make three times as likely to land on a problem involving both chapters 6 AND 7 than on a problem involving just one of the chapters, with all of the problems involving just one of the chapters being equally weighted. Find an appropriate probability model for this situation. Show your work please step by step.

d) Using the rigged die from part (c), find the probability that Alex will choose a problem which involves material from chapter 6 but not chapter 7. Show your work please step by step