First try:


The chances you would initially pick a fair coin are 3/4. The chances that a fair coin would flip two heads in a row is 1/4. So the chances that you picked a fair coin AND flipped 2 heads in a row with it is 3/4 * 1/4 = 3/16. Therefore the chance that you picked the double-headed coin is 13/16, since 3/16 plus 13/16 = 1.0


Intuitively this seems high, so I am going to think about it a bit more while I go out for a bit of exercise.