Coin equivalence: THT and TTH?

This is a follow-up to the question: http://answers.yahoo.com/question/index;_ylt=Ah6X0FC4qjh0w16E4QppFxPsy6IX;_ylv=3?qid=20080825143625AAdmSPo which is, in turn, a follow-up to http://answers.yahoo.com/question/index;_ylt=Anr2nrTJ85YJWm4CKDAmXIg8.Bd.;_ylv=3?qid=20080825135226AAIsU0t Suppose we flip a coin... show more This is a follow-up to the question:
http://answers.yahoo.com/question/index;...
which is, in turn, a follow-up to
http://answers.yahoo.com/question/index;...

Suppose we flip a coin until either the sequence THT or TTH shows up. If we want to have equal probabilities for either sequence to end our coin flipping, how should we weight the coin? That is, if the probability in any toss of heads is p, what value of p creates equal probabilities (of 1/2) of ending with THT or with TTH?

[I have two more follow-ups in mind; let me know if you like this enough to see some more...]
Update: You're approach is slightly shorter than mine, but we have the same conclusion. (If you don't start with an assumed T, you also got the "possibility" of p=0, which is likewise impossible.) I tried briefly to come up with a second question here that fixed this (i.e., made such a p possible) by... show more You're approach is slightly shorter than mine, but we have the same conclusion. (If you don't start with an assumed T, you also got the "possibility" of p=0, which is likewise impossible.)

I tried briefly to come up with a second question here that fixed this (i.e., made such a p possible) by adding a positive probability of the coin landing on its side, but that didn't work out. I'll give this question another day or so for any interesting insights, then go ahead and post the more difficult next question...
Update 2: Not that it really matters, but it irks me to see that I put "you're" at the beginning of that edit. Obviously that should be "your".
Update 3: Very good; here's the next one:
http://answers.yahoo.com/question/index?...
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