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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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    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?2012-05-05
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    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 2.2012-05-05
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    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?2012-05-05
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    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.2012-05-05
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    @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 $\frac{2}{3}\frac{9}{19}+\frac{1}{3}\frac{10}{19}=\frac{28}{57}$.2012-05-05
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    @AndréNicolas: Right you are. OP didn't say the chapters are in random order, but that is another way to see the difference.2012-05-05
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    @RossMillikan: He didn't in the main post, but did ask the question, third comment from the top, second sentence. I was just answering that, definitely not the question in the main post, which had been fully dealt with by your answer.2012-05-05
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    @AndréNicolas: Your logic makes it clear that there is a difference between (if there are two things, a and b) pick a random item-what is the chance of a and pick a random item and look at something else-what is the chance of a?2012-05-05
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    Thanks a lot friends! Just curious, where did $\frac{9}{19}$ and $\frac{10}{19}$ come from?2012-05-05
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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