20 August 2011

Fun With Epistemic and Aleatory Uncertainties

Deep in the comments on an earlier thread Paul Baer offers the following hypothetical:
In my statistics class, I ask my students "what is the probability that when I flip this coin, it will land heads." (And yes, assume it's a fair coin.)

Of course they answer 50% or some equivalent.

Then I flip it and hold it covered on the back of my hand. Then I ask, "What is the probability that this coin is heads." There's usually some puzzlement. Someone says "50%". And I say, "but either it's heads or it isn't. How can there be a fifty percent chance it's heads?"

Then I ask "what odds would you give me if I bet that it's not heads?" Eventually those who know what betting odds mean understand the point. Even when something has happened (like, the deck has been shuffled and the card that will be dealt could be known under some epistemic conditions DIFFERENT FROM OURS) we have to ACT as if the odds are, well, what we think they are.
To which I responded:
Consider the following case:

You flip a coin in your class and ask for the probability of a head. A savvy student replies:

[S1]: The odds of a head are 50-50

You then reveal to the class that the coin is not fair, in fact there is a 75% chance of a tail. You ask the student, now what are the odds of a head? (All while the flipped coin sits on your hand)

The student now replies:

[S2] The odds of a head are 25%.

Q1. Now would it be fair to say that [S1] was incorrect?