Austin, Bonnie, and Cathy are trying out for a school play. Assume each of them has a 50% chance of being cast in the play, and each student's audition is independent of the others. (a) List all possible outcomes for who will get a part in the play in the following table. Two outcomes have been filled in for you (there are many more.)
Outcome Austin Bonnie Cathy
1 YES YES YES
2 YES YES NO
Use the table to answer parts (b) - (e).
Remember each outcome is equally likely. (b) How many different outcomes are possible? (c) What is the probability that Austin will get a part, but Bonnie and Cathy will not? (d) What is the probability that exactly 2 of the students will get a part in the play? (e) If you know that exactly 2 students were cast, what is the probability that Bonnie was one of them? (Hint: Restrict your sample space in the table.) (f) Answer part (e) again using the formula for conditional probability. Your answer should match part (e). Show work.