A linearity fallacy is committed when you incorrectly assume two variables are related by a linear relationship. For example, assuming that adding people to a project decreases the time to finish proportionally[1] or that happiness scales linearly with income. This kind of fallacy, where you mistakenly draw an x-axis and a y-axis and a straight line going from the bottom-left to the top-right is a linearity fallacy that happens all the time.
But there's more. One day, you're walking along, and you decide to take two measurements of some property at different instances and get two real numbers, and . Numbers are numbers, right? Wait, before you go on, you should know that real numbers are... (read 1269 more words →)
Given the above formulation, the inference is not the same. P(H) = 1/2, but after information about W or not W is given, then P(H | W) = 1/3 or P(H | not W) = 1. The math doesn't care, you just aren't awake to perform your update process. When precommitting, you are not manipulating P(H), you are manipulating P(H |W) by changing W, so there's no issue.
P(A) = 1/2, P(B) = 0 is still the only way I can see to get P(H | W) = 1/2. In which case, I can't find any non-artificial framing for why Heads Tuesday does not exist (and Heads Monday exists twice as much as Tails Monday).
Wouldn't P(you wake up | heads) be 1/2, since you wake up on Monday but not on Tuesday? That would give P(you wake up) = 3/4, which gives P(heads | you wake up) = 1/3. That is, "you wake up" is not taken to mean "you wake up at least once," but "you wake up today (where today is Monday or Tuesday)." P(heads | you woke up at least once during the two days) is obviously 1/2.
I think not waking up/getting asked on HeadsTuesday (i.e. the Heads was flipped and the date is Tuesday) is important. This is the information you "gain" when you wake up (oh hey, it's not HeadsTuesday). When... (read more)