Five baseballs are thrown to a batter who attempts to hit the ball 350 feet or more. Let \(H\) denote the number of successes, with the \(p d\) for having \(h\) successes being \(f(h)=120 \times 0.4^{h} \times 0.6^{5-h} /[h !(5-h) !]\), where ! denotes the factorial operation.

a. Is \(H\) a discrete or continuous random variable? What values can it take?

b. Calculate the probabilities that the number of successes \(h=0,1,2,3,4\), and 5 . [Note: It may be convenient to use a spreadsheet or other software to carry out tedious calculations.] Sketch the \(p d f\).

c. What is the probability of two or fewer successes?

d. Find the expected value of the random variable \(H\). Show your work.

e. The prizes are \(\$ 1000\) for the first success, \(\$ 2000\) for the second success, \(\$ 3000\) for the third success, and so on. What is the \(p d f\) for the random variable PRIZE, which is the total prize winnings?

f. Find the expected value of total prize winnings, PRIZE.