### On the number of guns and planets out there, Part 2

(Note: It might take a while for the math symbols to load.)

In my previous post I set up the problem of gun statistics and planet statistics (where I mean math-problem, rather than trouble-problem). There's a question of the number of guns per capita, versus the fraction of the population with a gun (fraction of citizenry that are gun-owners). Also in there is the number of guns per gun-owner.

Similarly, there's the question of the number of planets per star throughout the galaxy, the fraction of stars with a planetary system, and the number of planets per system.

To illustrate this mathematically (and this involves nothing but multiplication and division, so stay with me!), let's set up two scenarios. Both scenarios have five stars and five planets:

Now let's introduce some mathematical terms. The first is the total number of stars in our sample, $N_\star$. Next is the number of planets, $n_p$. In both of the cases in the figure above, $n_p = 5$ and $N_\star = 5$. These two terms allow us to assess the number of planets per star:

$f_p = n_p/N_\star = 5/5 = 1$

So both cases give 1 planet per star. In terms of bulk numbers this is interesting. But it doesn't really give you a lay of the land. If you are searching for planets, the two cases pose very different situations with very different problems to solve. In Case 1, it's a hunter's bonanza: every target star has a planet to detect! In Case 2, it's 4 misses and one hit, but the one hit has a very complicated signal to decipher.

For planet-hunting, we want the fraction of stars with a planetary system $f_S$, where a system is one or more planets. Clearly the two example cases in the figure have very different values of $f_S$. This number is given by the number of systems $N_S$ divided by the number of stars $N_\star$. By inspection, Case 1 has $N_S = 5$ and $f_S=5/5=1$. Case 2 has $N_S = 1$ and $f_S = 1/5 = 0.2$.

A little more thought should reveal that $f_p$ and $f_S$ are related to one another. They're related by the average number of planets per system, given by $R$ (I'm running out of symbols!). To find the relation, let's figure out how one would use the various symbols above to come up with the number of planets in the sample. The number of planets is given by the number of stars, times the number of systems per star, times the number of planets per system:

$n_p = [{\rm Stars}] \times \left[\frac{{\rm Systems}}{\rm{Star}}\right]\times \left[\frac{{\rm Planets}}{\rm{System}}\right] = N_\star f_S R$
$\frac{n_p}{N_\star} = f_S R$
$f_p = R f_S$

Notice that if $R=1$, then $f_p = f_S$ because there is only one type of planetary system: those with a single planet. Of course, this isn't a likely scenario for planetary systems (just look at ours!), and it certainly isn't the case with most gun-owners. I'd hazard a guess that once you get to the point that you like guns enough to own one, you'll probably own several.

Also, note that $f_S$, or equivalently the fraction of gun-owners, must be less than or equal to one ($f_S \leq 1$). However, $f_p$ can be more than one, but for that to be the case, $R$ has to be bigger than one.

Anyway, the upshot is that the math is fairly simple, but the statistics of gun ownership and planet occurrence are somewhat subtle and you can get tied into knots unless your question is well-phrased. For the answer to the number of planets per star, stay tuned for our press release and final publication of our paper. For gun ownership, let's check out the numbers.

In the U.S. there are 89 guns per 100 civilians, or 0.89 per capita. We're number 1 in the world based on that statistic. (U-S-A! U-S-A! Suck it Serbia!). Thus $f_p = 0.89$.

But only 40-45% of adults own a gun (only!), or $f_S = 0.43$ to pick the average. This means that the number of guns per gun-owner is:

$R = f_p / f_s = 0.89/0.43 \approx 2$

This means that there are about 2 guns per gun-owner, on average.

So this is probably why Ta-Nehisi Coates had such a hard time squaring the various numbers for gun ownership. On the one hand, there are 0.89 guns per citizen. But only 45% of people own a gun, and when they do, they tend to have 2 guns.

### On the Height of J.J. Barea

Dallas Mavericks point guard J.J. Barea standing between two very tall people (from: Picassa user photoasisphoto).

Congrats to the Dallas Mavericks, who beat the Miami Heat tonight in game six to win the NBA championship.

Okay, with that out of the way, just how tall is the busy-footed Maverick point guard J.J. Barea? He's listed as 6-foot on NBA.com, but no one, not even the sports casters, believes that he can possibly be that tall. He looks like a super-fast Hobbit out there. But could that just be relative scaling, with him standing next to a bunch of extremely tall people? People on Yahoo! Answers think so---I know because I've been Google searching "J.J. Barea Height" for the past 15 minutes.

So I decided to find a photo and settle the issue once and for all.

I then used the basketball as my metric. Wikipedia states that an NBA basketball is 29.5 inches in circumfe…

### Finding Blissful Clarity by Tuning Out

It's been a minute since I've posted here. My last post was back in April, so it has actually been something like 193,000 minutes, but I like how the kids say "it's been a minute," so I'll stick with that.
As I've said before, I use this space to work out the truths in my life. Writing is a valuable way of taking the non-linear jumble of thoughts in my head and linearizing them by putting them down on the page. In short, writing helps me figure things out. However, logical thinking is not the only way of knowing the world. Another way is to recognize, listen to, and trust one's emotions. Yes, emotions are important for figuring things out.
Back in April, when I last posted here, my emotions were largely characterized by fear, sadness, anger, frustration, confusion and despair. I say largely, because this is what I was feeling on large scales; the world outside of my immediate influence. On smaller scales, where my wife, children and friends reside, I…

### The Force is strong with this one...

Last night we were reviewing multiplication tables with Owen. The family fired off doublets of numbers and Owen confidently multiplied away. In the middle of the review Owen stopped and said, "I noticed something. 2 times 2 is 4. If you subtract 1 it's 3. That's equal to taking 2 and adding 1, and then taking 2 and subtracting 1, and multiplying. So 1 times 3 is 2 times 2 minus 1."

I have to admit, that I didn't quite get it at first. I asked him to repeat with another number and he did with six: "6 times 6 is 36. 36 minus 1 is 35. That's the same as 6-1 times 6+1, which is 35."

Ummmmm....wait. Huh? Lemme see...oh. OH! WOW! Owen figured out

x^2 - 1 = (x - 1) (x +1)

So $6 \times 8 = 7 \times 7 - 1 = (7-1) (7+1) = 48$. That's actually pretty handy!

You can see it in the image above. Look at the elements perpendicular to the diagonal. There's 48 bracketing 49, 35 bracketing 36, etc... After a bit more thought we…