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:
- 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 $EX = n\times EX_1 = 1$
- Solving Buffon's needle by bending it. (AKA Buffon's noodle)
If you have any, please share them with me.