Sunday, December 27, 2015

One of two children is a boy, what's the probability the other is a boy?

It's one-half. Not one-third, one-half.

And no, none of that "both solutions work" or "it depends" or "the Bayesian or frequentist solution" nonsense. It's one-half for Bayesians, for frequentists, and for people who don't know what these categories mean. The only thing the one-third "solution" is good for is an example of the importance of understanding the difference between states of the world and observed events.

I go over the Bayesian solution and explain what's wrong with one-third in the following video:



I had a frequentist video, as well, but I deleted it in the big online cleaning of 2012, so here's a simple version.

There are four possible states of the world $\{(B_1,B_2),(G_1,B_2),(B_1,G_2),(G_1,G_2)\}$, where $B,G$ is the sex and the subscript is the birth order. From each of these states there are two possible events, observing the first or the second child, leading to eight possible state-event pairs:
\[\begin{array}{rlc}
(B_1,B_2) & \rightarrow & B_1 \\
(B_1,B_2) & \rightarrow & B_2 \\
(G_1,B_2) & \rightarrow & G_1 \\
(G_1,B_2) & \rightarrow & B_2 \\
(B_1,G_2) & \rightarrow & B_1 \\
(B_1,G_2) & \rightarrow & G_2 \\
(G_1,G_2) & \rightarrow & G_1 \\
(G_1,G_2) & \rightarrow & G_2
\end{array}\]
The event "one is a boy" means that only four of these state-event pairs are feasible in the universe of possibilities:
\[\begin{array}{rlc}
(B_1,B_2) & \rightarrow & B_1 \\
(B_1,B_2) & \rightarrow & B_2 \\
(G_1,B_2) & \rightarrow & B_2 \\
(B_1,G_2) & \rightarrow & B_1
\end{array}\]
The frequentist answer to "how likely is the state $(B_1,B_2)$?" is computed as the number of favorable pairs, that is pairs including $(B_1,B_2)$, two, divided by the total number of feasible pairs in the universe of possibilities, four. Two divided by four is one-half.

Why do people fall for the one-third "solution"? Two main reasons, I believe:

1. Understanding the difference between states and events and how to relate information to changes in probabilities is not a simple matter; most people think that they know how to do this better than they actually do.

2. The one-half solution sounds too simple, and therefore doesn't allow the person to affect sophistication. That, and most people's interest in STEM as an identity product only, is one of the most destructive mental attitudes one can have: it blocks learning.

Here's a different probability puzzle that suffers from the same problems:



In all fairness, in illo tempore when I saw the one-third "solution" I believed it correct, but my much smarter classmate Dave Godes immediately showed me it was wrong. To my credit, I didn't argue – when I'm shown to be wrong, I change my mind. I know, crazy.

- - - -
PS: Yes, I know that the frequentist computation is the total probability (not number) of the favorable pairs divided by the total probability of the feasible pairs. Since this is a simple example and all pairs are equiprobable, counting is equivalent to computing total probability in this computation.

Wednesday, December 23, 2015

Project 2016

Some acquaintances -- who never have time to learn anything new (or to exercise), although they seem to have vast knowledge of TV show storylines and sports events -- have challenged me to blog (their word was "prove") how it's possible to continue to learn stuff after formal education ends.

Hence "Project 2016," in which I'll mostly document the use of books, articles, MOOCs, podcasts, public lectures, and other sources for learning as entertainment.

Yes, learning as entertainment. I have to keep learning new things for my job; those will not be blogged. I like knowing stuff even when there's no monetary payoff to it. In the past I would keep the learning to myself, but since I like to peruse other people's educational blogs, I'll give something back to the community.

My main ludic learning interests are in STEM, economics, and business management. (I work on the quantitative side of business, but there's a lot beyond my area of expertise that I find enjoyable to learn.) There'll be references to books I read and I may occasionally stray into the application of logic and thinking to subjects like fitness, travel, or packing. I might even blog about science popularization.

There will be math (typesetting courtesy of MathJax):
\[
\Pr(\text{math}) = \lim_{x \rightarrow +\infty} \frac{x^{2016}-\log(x)}{x^{2016}}.
\]