Spoiler Alert:
Vg qrcraqf ba lbhe cevbe sbe gur vagryyvtrapr bs lbhe sevraq. Vs ur xabjf jung ur vf qbvat, ur jvyy perngr n jnecrq pbva gung vf fgvyy 50-50 naq lbh jvyy trg lbhe pybpx pyrnarq ertneqyrff bs nyy lbhe snapl Onlrfvna whwvgfh.
N havsbez qvfgevohgvba ba (0,1) vf whfg gur jebat cevbe sbe n thl gung pbzrf hc jvgu n jnecrq pbva naq n cebcbfvgvba sbe n org ba vg. Vg'f nyjnlf orggre gb juvc bhg n yvggyr tbbq frafr orsber lbh juvc bhg lbhe snapl rdhngvbaf.
V'z n yvggyr qvfnccbvagrq gung ab bar frrzf gb unir tbggra guvf lrg. Hfr lbhe urnqf.
V'z pbafvqrevat jurgure gb fuhg zl lnc naq znxr fbzr zbarl ng gur arkg ybpny YJ zrrgvat. ... V'z gbb ynml.
Presumably the rest of us are more interested in solving a fun math problem than in being smartasses. It is trivially easy to set up an example in which the uniform prior is valid. It so happens that the warped-coin story is more entertaining to think about, but before dismissing the problem on a technicality you should consider addressing the spirit of the problem rather than the letter.
I came up with this puzzle after reading Vaniver's excellent post on the Value of Information. I enjoyed working it out over Thanksgiving and thought I'd share it with the rest of you.
Your friend holds up a curiously warped coin. "Let's play a game," he says. "I've tampered with this quarter. It could come up all heads, all tails, or any value in between. I want you to predict a coin flip; if you get it right, I'll pay you $1, and if you're wrong, you pay me $3."
"Absolutely, on one condition," you reply. "We repeat this bet until I decide to stop or we finish N games."
What is the minimum value of N that lets you come out ahead on average?
Each game, you may choose heads or tails, or to end the sequence of bets with that coin. Assume that all heads:tails ratios are equally likely for the coin.
edit: since a couple people have gotten it, I'll link my solution: http://pastebin.com/XsEizNFL