Ok, I'm done. With this idea of gaining extra information (i.e. not like in the original pure probability riddle) I'm not sure we should talk about *the* probability that changing or keeping your door wins a certain fraction of all cases.
That's because now it depends on your strategy. For simplicity, let's consider my extreme hint example again where you *know* where the car is before your second decision. The probability for something is roughly speaking the good events divided by all events. Now if your strategy is "keep if and only if I was right" then you always win. That is, all your "change" decisions and all your "keep" decisions will win. Instead of 2/3 and 1/3 we now have 3/3 and 3/3. However, if your strategy is "keep if and only if I was wrong" then we drop to 0/3 and 0/3. So it depends on your strategy and thus speaking about *the* probability for keep/change doesn't make much sense, at least not directly. I guess this term could make sense if you define it over *all strategies*, and then it might be 2/3 and 1/3 again, but I won't dare to claim that cause now it's getting too complicated :-) If we talk about keep/change as strategies, short for "I'll always keep my door" (or "change"), then 2/3 and 1/3 is correct. It's also correct if your choice is random. Or if you always alternate between keep/change (unless the correct door alternates accordingly). Or other constructions that keep the 2/3 vs 1/3 probabilities for keep vs change. But inside other strategies, the probabilities can be very very different. Cheers! Stefan --- In [email protected], "Stefan Pochmann" <[EMAIL PROTECTED]> wrote: > > --- In [email protected], Rune Wesström <rune. > [EMAIL PROTECTED]> wrote: > > > > "Good thinking Stefan!" ? He is contradicting himself when saying: > "you can gain some knowledge this way and be more sure that changing > would be better...but... the probability stays at 2/3". But if we had > a probability of 2/3 whithout this knowledge and then bettered it, how > can we still stay at 2/3? > > > Hehe, you better don't remove the important parts of the quote :-). I > was talking about two different things. I said: > > > > Yes, you can gain some knowledge > > > this way and be more sure that changing would be better. But it > > > doesn't make you win more often when changing. The probability for > > > that stays at 2/3. > > The probability that "change" wins the car stays at 2/3 and the > probability for "keep" stays at 1/3. The extra knowledge doesn't help > to make these strategies any better, "keep" wins if and only if your > initial choice was correct (1/3) and "change" wins if and only if it > was wrong (2/3). > > What you *can* do with this extra knowledge is make up a *better > strategy* than those two. In my extreme example where you get to see > the car, a good strategy would be "keep my door if and only if it > contains the car". This *new* strategy wins the car 100% of the time. > > Now what about the situation you described? The game is the same but > you get extra information from the behaviour of the host. Can you use > it to make up a new strategy? > > Hmmmmm.... actually I just noticed something. Gotta think about it, > will be back when I'm done with that. But I'm gonna submit the above > now anyway. > > Cheers! > Stefan > ------------------------ Yahoo! Groups Sponsor --------------------~--> Get fast access to your favorite Yahoo! Groups. Make Yahoo! your home page http://us.click.yahoo.com/dpRU5A/wUILAA/yQLSAA/MXMplB/TM --------------------------------------------------------------------~-> Yahoo! Groups Links <*> To visit your group on the web, go to: http://groups.yahoo.com/group/speedsolvingrubikscube/ <*> To unsubscribe from this group, send an email to: [EMAIL PROTECTED] <*> Your use of Yahoo! Groups is subject to: http://docs.yahoo.com/info/terms/
