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I am going to give a presentation about the basics of probability and for that, I would like to include some interesting (or even better mind-blowing) examples of how linearity of expectation can help solve problems. I have two examples in mind for now, but before proceeding, I thought it was good to ask for other ideas.

My examples are:

  1. Expected value of the number of fixed points in a random permutation.

You can write $X = X_1 + X_2 + \ldots + X_n$ , and then $EX = n\times EX_1 = 1$

  1. Solving Buffon's needle by bending it. (AKA Buffon's noodle)

If you have any, please share them with me.

MR_BD
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    This is very well-documented already: https://math.stackexchange.com/questions/3985462/striking-applications-of-linearity-of-expectation#:~:text=Linearity%20of%20expectation%20is%20a,otherwise%20make%20any%20calculation%20daunting. – Danny Duberstein Sep 10 '23 at 10:28

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