This delightful problem is courtesy of Diana Davis (don’t follow the link unless you want to see the solution, though!), although I did take the liberty of formulating it slightly differently. =)
One bright and sunny day, you’re walking down the street thinking about fractals, and you are accosted by a strange-looking woman with clothes covered in printed triangles, protractors, and compasses (a la Ms. Frizzle from the Magic School Bus). “I have a mathematical proposition to make!” she says. Your initial reaction is to walk right by mumbling something about thanks but I already have a protractor… but something makes you pause to listen instead. “We will pick a random triangle,” she continues, “and if it’s obtuse, you pay me $1, but if it’s acute, I’ll pay you $2!” “Hmm…” you say, “what do you mean by a ‘random triangle’?” “Oh, we will just roll these two special dice here,” she replies. “They have infinity sides, and when you roll them, they give you a completely random number between 0 and 180 — like 28.3, 94.11106749…, and so on. We’ll use them to pick two of the angles, and of course, we can figure out the other angle since all the angles of a triangle have to add up to 180 degrees.” She rolls her special dice a few times to show you, and although you can’t quite understand how dice could have infinity sides, they do seem to give random numbers between 0 and 180 as she claims.
Should you take the bet?