A book is $90\%$ likely to be illustrated; an illustration is $90\%$ likely to be in color. How many of $10000$ books must have a color illustration?

Question

This problem is in Art and Craft of Problem Solving by Paul Zeitz. It's an easy looking puzzle which is not "so obvious" according to the problem itself. It says -

Of all the books at a certain library, if you select one at random, then there is a $90\%$ chance that it has illustrations. Of all the illustrations in the book, if you select one at random, then there is a $90\%$ chance that it is in color. If the library has $10000$ books, then what is the minimum number of books that must contain colored illustrations?

I immediately got an answer, but the warning in the problem makes me thoughtful. Isn't it obvious, you have $\frac{90}{100}\cdot 10000=9000$ books with illustrations and then $\frac{90}{100}\cdot 9000=8100$ books with colored illustrations? Or am missing something obvious?

If, in each of the 9000 books with illustrations, you have a $90%$ chance of the illustration you pick being in color, then each of those 9000 books must have color illustrations. — Joe, Apr 18 '20 at 10:55

score 4 · Accepted Answer · answered Apr 18 '20 at 20:57

4

I am the author of this book. The problem as is misquoted. The correct language for the second sentence begins, "Of all the illustrations in all the books..." (cf. p. 25 of the second edition).

The problem is tricky, but the unambiguous answer is "one," since it is possible that one of the books in the library has the title, "the big book of illustrations, including all colored images in the library."

answered Apr 18 '20 at 20:57

Paul Zeitz

56

Thats a tricky and very funny one! Nice to see you here :) – Apr 19 '20 at 01:44
I love your book! Great puzzles and good practice. – Robin Aldabanx May 02 '20 at 00:40

score 3 · Answer 2 · answered Apr 18 '20 at 10:58

I suspect there is an error in the question as it doesn't make sense at the moment. "Of all the illustrations in the book..." - which book?

I suspect that is intended to say "Of all the illustrations in the library...", with the idea being that every book either has coloured illustrations, uncoloured illustrations, or no illustrations. The trick is that there is a big difference between choosing a random illustration, and choosing a random illustrated book.

In this case, there could potentially be only one book with coloured illustrations, which contains $90\%$ of all illustrations in the library. Since this book must have at least $80991$ illustrations (there are $8999$ other books with at least one illustration each), this is unfeasible but not impossible.

It definitely is ambiguously worded. I took it to mean, given that you chose a book with illustrations, the probability that the illustration you choose in “the book” is color is $90%$. I believed that it had to be a conditional probability, since if you chose a book without illustrations, it doesn’t make sense to say the probability that the illustration you choose in “the book” is color is $90%$. — Joe, Apr 18 '20 at 11:08

A book is $90\%$ likely to be illustrated; an illustration is $90\%$ likely to be in color. How many of $10000$ books must have a color illustration?

2 Answers2