Two professors at a nearby university want to co$-$author a new textbook in either economics or statistics. They feel that if they write an economics book, they have a $50$ percent chance of placing it with a major publisher, and it should ultimately sell about $40,000$ copies. If they can't get a major publisher to take it$,$ then they feel they have an $80$ percent chance of placing it with a smaller publisher, with ultimate sales of $30,000$ copies. On the other hand, if they write a statistics book, they feel they have a $40$ percent chance of placing it with a major publisher, and it should result in ultimate sales of about $50,000$ copies. If they can't get a major publisher to take it$,$ they feel they have a $50$ percent chance of placing it with a smaller publisher, with ultimate sales of $35,000$ copies.

a$.$ Construct a decision tree for this problem.

b$.$ What is the probability that the economics book would wind up being placed with a smaller publisher?

c$.$ What is the probability that the statistics book would wind up being placed with a smaller publisher?

d$.$ What is the expected value for the decision alternative to write the economics book?

e$.$ What is the expected value for the decision alternative to write the statistics book?

f$.$ Which decision alternative should be selected and what is the expected value for this $($the optimum$)$

alternative?