Joan will grade four statistics projects, selected at random from a large stack, this evening. From past experience, Joan thinks that 30% of the projects will be "A" quality projects. Let X be the number of "A" projects she grades tonight. (Assume that all the projects are independent and that Joan is completely fair and objective with her grading of each one.)
a. Find the probability mass function.
b. Is this a valid probability mass function? Justify your answer.
c. What is the probability that Joan grades at least two "A" projects tonight?
d. What is the probability that Joan grades at most two "A" projects tonight?
e. What is the probability that Joan grades exactly four "A" projects given that she grades at least two "A" projects tonight?
f. What is the probability that Joan grades at most four "A" projects given that she grades strictly more than two "A" projects tonight?