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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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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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    @DavidFaux: If you leave a chapter, there are 19 chapters that could be the next one. 9 of them are the length of the one you are in (because the next one can't be the one you are in) and 10 are the other length. So it is $\frac 9{19}$ that the chapter after a long one is long.2012-05-05