## Monday, November 12, 2012

Q: Sleeping Beauty (SB) has volunteered for a test. The test is structured as follows. On a Sunday SB is put to sleep and after that a coin is tossed. If the coin
1. Comes up HEADS, SB is awoken on the following day (Monday) and asked her belief in the probability that the toss came up heads. The experiment ends.
2. Comes up TAILS, same as above except that at the end of the question her memory of the waking event and question is wiped out and she is awoken the next day (Tuesday) and asked the same question. The experiment ends.
The question is, when SB is awoken, what is SB's belief in the probability that it was heads. Note: At any given time, SB has no way to tell if it is a Monday or a Tuesday.

A:  While it is tempting (and there are some arguments out there) to say the probability is $$\frac{1}{2}$$, the reality isn't so. Here is why. Take a look at the following table.

The "X" indicates the situations when SB wakes up. Given that SB does not know which day of the week it is, she would guess $$P(Heads) = \frac{1}{3}$$. If she knew that the day was a Monday, then we need only consider the Monday column and $$P(Heads) = \frac{1}{2}$$. Likewise, if she knew it was a Tuesday then $$P(Tails) = 1$$.

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Here are a few
Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.

Introduction to Algorithms
This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists

Introduction to Probability Theory

An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)

Introduction to Probability, 2nd Edition

The Mathematics of Poker
Good read. Overall Poker/Blackjack type card games are a good way to get introduced to probability theory

Let There Be Range!: Crushing SSNL/MSNL No-Limit Hold'em Games
Easily the most expensive book out there. So if the item above piques your interest and you want to go pro, go for it.

Quantum Poker
Well written and easy to read mathematics. For the Poker beginner.

Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
An excellent resource (students/engineers/entrepreneurs) if you are looking for some code that you can take and implement directly on the job.

Understanding Probability: Chance Rules in Everyday Life A bit pricy when compared to the first one, but I like the look and feel of the text used. It is simple to read and understand which is vital especially if you are trying to get into the subject

Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems) This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

#### 1 comment:

1. While the answer is indeed 1/3, your argument is (barely) not right. It requires one more trivial step. But the halfers, those who favor the 1/2 answer, will simply dismiss it citing that step as the reason. They usually dismiss it with that step as well, but they never say why. Because they can't.

Instead of finding a flaw in the more explicit thirder reasoning, halfers claim that SB has no "new information" on which to base a change in the probability. But they never define what actually is necessary to do that, and it turns out you do have it.

Anyway, the original argument is that IF you are told it is Monday, you should have equal confidence that the coin landed heads or tails. This means that P(Heads&Monday)=P(Tails&Monday). Similarly, IF you are told that the coin landed tails, you should have equal confidence that it is Monday or Tuesday. This means that P(Tails&Monday)=P(Tails&Tuesday). Since I've listed the three exclusive possibilities, and they must be equal, it follows that all equal to 1/3.

The argument that directly contradicts the halfers is that, AT ANY TIME DURING THE EXPERIMENT, there are four possible combinations of day and coin, not three. Each has a prior probability to apply to a random moment of 1/4. Since she is awake, SB knows that the combination Tuesday&Heads is ruled out, so the updated probability for each of the other three is (1/4)/(1/4+1/4+1/4)=1/3.