Question

An author has written a book and submitted it to a publisher. The publisher offers to print the book and gives the author the choice between a flat payment of $$\$ 10,000$$ and a royalty plan. Under the royalty plan the author would receive $$\$ 1$$ for each copy of the book sold. The author thinks that the following table gives the probability distribution of the variable \(x=\) the number of books that will be sold: $$ \begin{array}{lrrrr} x & 1000 & 5000 & 10,000 & 20,000 \\ p(x) & .05 & .30 & .40 & .25 \end{array} $$ Which payment plan should the author choose? Why?

Short answer

The author should choose the royalty plan because the mathematical expectation of $13,250 is greater than the flat payment of $10,000. While there is a risk in the royalty plan, the potential benefits outweigh the risk.

Step-by-step solution

Understanding the Payment Plans

The book author has two choices: a) a flat payment of $10,000 or b) a royalty of $1 for each book sold. Therefore, for the royalty plan the total earning will be equivalent to the number of books sold. The probabilities given for each number of books sold will be used to calculate mathematical expectation.

Calculating the Mathematical Expectation for the Royalty Plan

The mathematical expectation is the sum of possible earnings multiplied by their corresponding probabilities. So, it will be \(E = 1000*1*0.05+5000*1*0.3+10000*1*0.4+20000*1*0.25 = 13250\). Therefore, the expected earnings from the royalty plan are $13,250.

Comparing both Plans

The flat payment plan offers $10,000 irrespective of the number of books sold. The royalty netted an expected earning of $13,250. Both values are compared and analysed under the fact that the royalty plan holds a risk, as there is a probability of selling a smaller number of books.