Be Afraid, Be Very Very Afraid

Two for the price of one today:

1)  First, in time for Halloween, DO NOT miss the frightful tale of Differentiation... as only Ben Orlin can tell it (bwaaahaaaahaaaaa):
http://mathwithbaddrawings.com/2015/10/28/the-differentiation-a-survivors-tale/


2)  Less scary, but more mind-racking perhaps than differentiation, is the 'Sleeping Beauty Problem/Paradox,' which I haven't mentioned for awhile, but do now (...at least one version of it):



The correct answer is: 1/2, 1/3?; 1/2, 1/3?; 1/2 or 1/3???.... two different logical answers, splitting the mind in two, with no final resolution. Spine-tingling stuff! ;-)
My original post on it was back in 2012 with a number of additional links:
http://math-frolic.blogspot.com/2012/03/sleeping-beautynot-your-childhood-fairy.html

Also, Tanya Khovanova had lengthy previous discussion of it on her blog here:
http://blog.tanyakhovanova.com/2011/08/the-sleeping-beauty-problem/

And even physicist Sean Carroll covered it a year ago, drawing 240+ comments:
http://www.preposterousuniverse.com/blog/2014/07/28/quantum-sleeping-beauty-and-the-multiverse/

Pick your side... you'll find some good arguments (and thinkers) backing you up either way.  Spooky indeed!

Re-visiting Sleeping Beauty

(via Rachel CALMUSA/WikimediaCommons)

In chapter 11 ("Is Time an Illusion") of Max Tegmark's new book "The Mathematical Universe" the author, while discussing the nature of time and human consciousness, touches upon the "Sleeping Beauty" puzzle/paradox, which I mentioned here almost two years ago:

http://math-frolic.blogspot.com/2012/03/sleeping-beautynot-your-childhood-fairy.html

This is one of the most interesting and delicious (perhaps even complicated, in some ways) puzzles around, as people argue vociferously for either of two different answers (1/2 or 1/3), because of the conditional probabilities involved.

[Here is one statement of the puzzle: Sleeping Beauty undergoes the following experiment, being told all these details ahead of time. On Sunday she will be put to sleep. A fair coin will then be tossed to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday AND Tuesday. But when she is put to sleep again on Monday, she is given an amnesia-causing drug which ensures she cannot remember the prior awakening. In this case, the experiment ends after she is interviewed on Tuesday.  Whenever Sleeping Beauty is awakened and interviewed, she is asked, "What do you believe is the probability that the tossed coin landed on  heads?" -- What is her answer?]

In re-visiting the links I provided in my original blog post I discovered that the "Tanya Khovanova" link has since added further lo-o-ong discussion of the issues by two commenters back-and-forth, which is probably worth checking out if you are especially interested in probability in general, or this problem in particular (if these areas don't interest you, don't visit it, lest you fall into a deep, deep coma, or alternatively, your head explode ;-)

Sleeping Beauty...NOT Your Childhood Fairy Tale

I've been reading about the "Sleeping Beauty Problem (or Paradox)" lately. It's actually a decade-plus-old quandary that I was aware of, but had never paid much attention to until it popped up somewhere on one of my Twitter feeds last week. Loving a good paradox, it's been rattling in my brain since.
Some folks say it reminds them of "The Monty Hall Problem" in so much as people argue vigorously for different solutions. But the Monty Hall Problem has an actual correct answer, whereas (so far as I can tell) the SBP really can be argued in two different, divergent approaches (designated as "halfers" and "thirders"). The paradox is sometimes stated in slightly variable ways, which is part of the problem, but even a fairly standard statement of it can include slight semantic pitfalls, leading to some of the disagreement. Still, more than 'Monty Hall,' the SB problem reminds me of the famous "Newcomb's Paradox" where people also tend to split two ways, and there simply is no established "right" answer.

Wikipedia states the SB problem as follows:

"Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details. On Sunday she is put to sleep. A fair coin is then tossed to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday and Tuesday. But when she is put to sleep again on Monday, she is given a dose of an amnesia-inducing drug that ensures she cannot remember her previous awakening. In this case, the experiment ends after she is interviewed on Tuesday.

Any time Sleeping beauty is awakened and interviewed, she is asked, "What is your credence now for the proposition that the coin landed heads?"

Many re-state the problem to ask Sleeping Beauty, "What is the probability now for the proposition that the coin landed heads?," and some argue this word change is significant and alters the discussion. I think most people however, indeed understand the problem in terms of probabilities (the probability of heads being either 1/3 or 1/2), and so I certainly prefer thinking about it that way.

The Wikipedia entry for the problem is here:

http://en.wikipedia.org/wiki/Sleeping_Beauty_problem

And here are some more links discussing it (warning though, they may cause your head to pound ;-)):

http://blog.tanyakhovanova.com/?p=356 (arguing for the 1/3 answer)

http://barryispuzzled.com/zbeauty.htm (arguing for the 1/2 answer)

http://meteuphoric.wordpress.com/2011/01/09/against-the-hybrid/

…and discussion from a "freethought forum" here:

http://www.freethought-forum.com/forum/showthread.php?t=3120

Finally, if that's not enough for you, you glutton-for-punishment, more links here:

http://www.anthropic-principle.com/?q=resources/preprints#pre5