Pickwick's Umbrella

I wrote this in 2010, and it's been copied a bit - seem to find it here and there.  I saw the problem originally in the chance probability document which is still available  (https://chance.dartmouth.edu/teaching_aids/books_articles/probability_book/book.html),
with exercise answers --  and the answer to this problem was wrong - so I contacted the authors, and had some fun.