Sure. By tweaking your “weights” or other fudge factors, you can get the right answer using any probability you please. But you’re not using a generally-applicable method, that actually tells you what the right answer is. So it’s a pointless exercise that sheds no light on how to correctly use probability in real problems.
Completely agree. The general applicable method is:
Understand what probability experiment is going on, based on the description of the problem.
Construct the sample space from mutually exclusive outcomes of this experiment
Construct the event space based on the sample space, such that it was minimal and sufficient to capture all the events that the participant of the experiment can observe
Define probability as a measure function over the event space, such that:
The sum of probabilities of events consisting of only individual mutually exclusive and collectively exshaustive outcomes was equal to 1 and
if an event has probability 1/a then this event happens on average N/a times on a repetition of probability experiment N times for any large N.
Naturally, this produce answer 1⁄2 for the Sleeping Beauty problem.
If Beauty thinks the probability of Heads is 1⁄2, she presumably thinks the probability that it is Monday is (1/2)+(1/2)*(1/2)=3/4
This is a description of Lewisian Halfism reasoning, that in incorrect for the Sleeping Beauty problem
I describe the way the Beauty is actually supposed to reason about betting scheme on a particular day here.
She needs a real probability.
Indeed. And real probability domain of function is event space, consisting of properly defined events for the probability experiment. “Today is Monday” is ill-defined in the Sleeping Beauty setting. Therefore it can’t have probability.
Completely agree. The general applicable method is:
Understand what probability experiment is going on, based on the description of the problem.
Construct the sample space from mutually exclusive outcomes of this experiment
Construct the event space based on the sample space, such that it was minimal and sufficient to capture all the events that the participant of the experiment can observe
Define probability as a measure function over the event space, such that:
The sum of probabilities of events consisting of only individual mutually exclusive and collectively exshaustive outcomes was equal to 1 and
if an event has probability 1/a then this event happens on average N/a times on a repetition of probability experiment N times for any large N.
Naturally, this produce answer 1⁄2 for the Sleeping Beauty problem.
This is a description of Lewisian Halfism reasoning, that in incorrect for the Sleeping Beauty problem
I describe the way the Beauty is actually supposed to reason about betting scheme on a particular day here.
Indeed. And real probability domain of function is event space, consisting of properly defined events for the probability experiment. “Today is Monday” is ill-defined in the Sleeping Beauty setting. Therefore it can’t have probability.