## § Counter-intuitive linearity of expectation [TODO ]

• I like the example of "10 diners check 10 hats. After dinner they are given the hats back at random." Each diner has a 1/10 chance of getting their own hat back, so by linearity of expectation, the expected number of diners who get the correct hat is 1.
• Finding the expected value is super easy. But calculating any of the individual probabilities (other than the 8, 9 or 10 correct hats cases) is really annoying and difficult!
• Imagine you have 10 dots scattered on a plane. Prove it's always possible to cover all dots with disks of unit radius, without overlap between the disks. (This isn't as trivial as it sounds, in fact there are configurations of 45 points that cannot be covered by disjoint unit disks.)
• Proof: Consider a repeating honeycomb pattern of infinitely many disks. Such a pattern covers pi / (2 sqrt(3)) ~= 90.69% of the plane, and the disks are clearly disjoint. If we throw such a pattern randomly on the plane, any dot has a 0.9069 chance of being covered, so the expectation value of the total number of dots being covered is 9.069. This is larger than 9, so there must be a packing which covers all 10 dots.