So, my options are guessing that the rigged die will land on 1, or <2,3,4,5,6>.
I do not know which number it is rigged to land on.
Therefore I cannot make any assumptions about which number it is rigged to land on.
In reality, the probability of the number it lands on is:
rigged number=1
other numbers=0.
But to me, the probability of each number being the rigged number is 1/6. So adding that fact into my calculation does nothing to influence my decision. All I know is, guessing that it will land on a number higher than 1 contains 5 out of the 6 possible outcomes, while guessing 1 contains 1 out of 6 possible outcomes. Therefore it is more reasonable.
How is this any different from simple probability?
Probability does not hinge on truth; that is, the fact that it may be rigged to land on 1 in no way undercuts the rationale that given what I know, is it more probable to guess not-1 than 1.
It is only because it is more probable to guess not 1 that makes it "more reasonable". The fact that the dice is rigged is negated by the fact that I do not know which number it is rigged to land on. Thus each number has 1/5 chance of being the rigged number. So there is no point introducing either element. You could have simply asked me to guess 1 or not-1 with a fair die. The analysis is exactly the same.
I take from this that you think it is more reasonable to predict the dice will land on a number higher than one.
ReplyDeletethe faces probabilities are unknown.
the probabilities concerning how many times an observed pattern will continue are also unknown.
Because, in the dice example, there are more possibilities in the group "higher than one", it is more reasonable to choose that group over "one".
There are more possibilities in the group "the pattern continues at least once" so it is reasonable to choose that group over "the pattern doesn't continue"
Just because the ACTUAL probabilities are unknown, that does not rule out using probability to pick the option with more possibilities as you say.
ReplyDeleteProbability can go beyond actual, it can be circumstantial, uninformed, biased, etc. I still suspect that because you consider the probabilities to be unknown given the situation, that you must use 'more reasonable' instead. I'm not convinced this is the case, I think the choice of 'not 1' is informed by a probabilistic premise such as:
"if all values are equal, choose the option with the most possibilities over those with less possibilities"
Or something to that effect. Therefore I think you may as well say 'not 1' is more probable, given the circumstances.
"Just because the ACTUAL probabilities are unknown, that does not rule out using probability to pick the option with more possibilities as you say."
ReplyDeleteah, yup. cant disagree with that. infact thats about the crux of my argument.
"if all values are equal, choose the option with the most possibilities over those with less possibilities"
i dice example the values (i assume you mean probabistic value) are not equal, their unknown.
"Therefore I think you may as well say 'not 1' is more probable, given the circumstances"
your saying “not one” is more probable? i agree.
and of corse to pick the option which is "more probable" is going to be "more reasonable" which is why i use both terms interchangably.