|As I did say at the start, the answer was 1/3 and I have changed the wording, searching for a less ambiguous wording.|
The answer to the problem in the OP is certainly not 1/3. It's 1/2.
The answer to this problem:
|What's the probability of HH given that ¬TT?|
1/3. They sound very much alike, but they're entirely different. ¬TT gives a lot
of information about the system. In the original problem, there was some uncertainty about the exact state the system was in because when you looked at the coins and said "not double tails" or "not double heads" there was some overlap at HT and TH. Without that overlap, the uncertainty is gone, and the probability does become 1/3:
P(HH | ¬TT) = P(¬TT | HH)P(HH) / (P(¬TT | HH)P(HH) + P(¬TT | HT)P(HT) + P(¬TT | TH)P(TH) + P(¬TT | TT)P(TT)) =
= P(¬TT | HH) / (P(¬TT | HH) + P(¬TT | HT) + P(¬TT | TH) + P(¬TT | TT)) =
= 1 / (1 + 1 + 1 + 0) =
Which is obvious. If it's not one of the four things, it has to be one of the other three, which still have uniform distribution.
|I think HH' means "anything other than heads-heads". Is that true? If so, I think P(HH') is 3/4. If it's not true, what does HH' mean?|
Like I said, this was the first thing I defined:
|Let HH, TH, HT, and TT be the coin configurations, and TT' be you saying "it's not double tails".|
My definition implies that even if the coins land on HT or TH, you may or may not say "it's not double tails".