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A book has 10 short and 10 long chapters. Short chapters span 10 pages, and long chapters span 20 pages.

Why does the probability that you will pick a long or a short chapter differ between these strategies?

Strategy #1: Flip to a random page, back up to the start of that chapter, and start reading.
Strategy #2: Flip to a random page, go forward to the start of the next chapter, and start reading (and pick the first chapter if the page you pick lies within the last chapter).

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up vote 4 down vote accepted

The book has 300 pages, 200 of which are in long chapters and 100 of which are in short chapters. If you pick a random page it is $\frac 23$ to be in a long chapter. So strategy 1 gives you a long chapter $\frac 23$ of the time. Strategy 2 gives you the chapter after a long one $\frac 23$ of the time. If the chapters alternate, strategy 2 will give you a short chapter $\frac 23$ of the time.

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Thanks, but doesn't the book still have 200 pages from long and 100 pages from short chapters both ways? How come the probability of picking long chapters isn't $\frac{2}{3}$ for both strategies? In other words, doesn't that logic for strategy 1 also also apply to 2? – David Faux May 5 '12 at 0:04
The initial page selection is in fact $\frac 23$ to be in a long chapter either way. But the selected chapter in strategy 2 is not the chapter the selected page is in. Think of a 2 chapter book with the first chapter 200 pages and the second 100. $\frac 23$ of the time the page chosen will be in chapter 1. In strategy 1 you then read chapter 1. In strategy 2 you then read chapter 2. – Ross Millikan May 5 '12 at 0:08
Ah thanks. So if short and long chapters alternate, then a reader using strategy 2 flips forward to a short chapter if he/she lands in a long chapter. However, what if the chapters don't alternate and are ordered randomly? Would the probability of reading a short chapter for strategy 2 be less than $\frac{2}{3}$ because the reader will not always move forward to a short chapter after landing in a long chapter? – David Faux May 5 '12 at 0:23
It will depend upon the ordering of the chapters. If chapters 1-5 are long and 2-6 are short, you will have $\frac 2{15}$ to read any of 2-6 and $\frac 1{15}$ to read 1 or 7-10. So that is $\frac 9{15}$ to read a long and $\frac 6{15}$ to read a short. – Ross Millikan May 5 '12 at 0:34
@DavidFaux: If the chapters are in random order, and you land in a long chapter, the probability the next one is long is $\frac{9}{19}$, while if you land in a short, probability next is long is $\frac{10}{19}$. So the probability you end up in a long with second strategy and random chapter order is $\frac{2}{3}\frac{9}{19}+\frac{1}{3}\frac{10}{19}=\frac{28}{57}$. – André Nicolas May 5 '12 at 2:04

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