Question

A retail giant, Mississippi, sells books on its website. The book Lisa Panner and the Accountant’s Ledger (LP for short) is sold in the hardcover version, paperback version, audiobook version, or the e-book version to be read on its proprietary device Ignite. It is known that no one buys both hardcover and paperback of LP, and no one buys both hardcover and e-book. It is also known that no customer buys all three of paperback, e-book and audiobook. Of the customers interested in LP, Mississippi has found that 15% have bought the hardcover, 35% have bought the paperback, 30% have bought the audiobook and 25% have bought the e-book. Furthermore, 40% have bought either the hardcover or the audiobook, and 10% have bought both the paperback and the audiobook.

(a) What fraction of the interested customers have bought both the hardcover and the audiobook?

(b) What fraction of the interested customers have bought exactly one of the paperback or the audiobook?

(c) (Somewhat hard) If 5% of interested customers have bought both audiobook and e-book, what fraction have bought only the audiobook? (Hint: Drawing the Venn diagram will be helpful. Try to break the set of customers who bought an audiobook into mutually exclusive sets so that you can use additivity.)

Answer 1

Let H represent the hardcover version of LP books sold

A represent the audiobook version of LP books sold

E represent the e-book version of LP books sold

P represent the paperback version of LP books sold

n(*) represent the fraction of *

Given verbatim data in symbol form:

n(H) = 15/100 = 0.15, n(P) = 0.35, n(A) = 0.30, n(E) = 0.25

a) Fraction of the customers who have bought both the hardcover and the audiobook versions =

Addition rule:

Therefore the fraction of the customers who have bought both the hardcover and the audiobook versions is 0.05

b) Fraction of the customers who have bought exactly one of the paperback or the audiobook version =

Addition rule:

Therefore the fraction of the customers who have bought exactly one of the paperback or the audiobook version is 0.45

c) The following venn diagram shows the required area,

From the venn diagram, fraction of the customers who have bought only the audiobook version =

Given

Also from (a)

Therefore the fraction of the customers who have bought only the audiobook version is 0.30