## Friday, March 10, 2017

### Sleeping Beauty is Easy

Some very clever arguments for making it much harder than it actually is.

The Monty Hall problem is easy if you simply write out the possibility tree and count the final nodes. If I get fancier than counting, it gets tricky. I got this one wrong too before I checked using the dumb counting method.
Though there's a trick depending on what I care about.

The options above the line are 50% each. All I can do is change the weightings of the possibilities below the line, which must also sum to 50%.
The brain gets confused because there's more Ts in the lower half. However, since they must sum to 50%, they're worth less. More on this at the bottom.

If you run the experiment once, then it is kind of hard to think about. Instead, run it a thousand times.
If I think the probability of tails is 1/2, I will report it half the time, and my probability will match the number of times I'm right.
If I report tails all the time, I will be right half the time. If I report heads all the time, I'll be right half the time.
If I think the probability of tails 2/3rds, I will report it more often in the bottom, but less often than in the top, and be right half the time. (1/9 + 1/9 + 4/9 = 6/9 = 2/3. 2/6 + 1/6 = 1/2.)

I dunno. Doesn't seem like I need a whole big book of calculus to solve the problem. I just have to put aside my pride long enough to do the dumb counting thing.

Self-location information? Conditionalization? Principle of indifference? Imagine rule? Impressive, clever-sounding things I don't need to know or care about. I just make a stupid picture and look at it.

The trick being my choice of perspective. If I want to be right, I can pick whatever I want, as I'll be right half the time. If instead we're talking about the number of reports the experimenter sees, then it's different. If the observer is writing down each report separately, and I want to generate as many 'matching' entries as possible, it favours reporting tails. Tails causes double the records, effectively doing the experiment twice on that branch.