was recently asked how I might go about working out the expectation value of the game where an unbiased coin is thrown n times where you win nothing each time a head is thrown and you win $n if you throw a tail on the nth throw.
I hadn’t thought of this problem for a long time and didn’t have a pen and paper on hand and couldn’t give an answer (!) - but when I got home, quickly playing with the problem with pen and paper showed that, of course, the expectation value is
$n(n + 1)/4.
Pen and paper – the tools of a mathematician! It would be nice to have a mental whiteboard on which to do calculations, but paper is much more reliable!