Your Complete Guide To Betting On Interplanetary Birthdays

Help!

I’ve fallen down a rabbit hole at 3.711 m/s² and I can’t get out…

A few weeks ago, I sang the praises of one of my favorite probabilistic puzzles, the Birthday Problem. As a refresher, this asks the question: “In a group of people, are there any Birthday Twins who happen to share the same birthday?”

An interesting alternative way to frame the problem is to gamble on whether or not there are any Birthday Twins at your party. In this case you would try to figure out how many people you need in order to have a 50-50 chance of finding some “B-Twins” amongst you. Then, if you have less Party People than that Magic Number, you bet against B-Twins, and bet on them if your number of Party People is the Magic Number or higher.

Now (as previously shown) the Magic Number is normally 23…

I say “normally” because this is based on the reasonable assumption that you’re on Earth, exclusively amongst native-born Earthlings.

But…but, what if…?

When I originally brought up the Birthday Problem, I thought it would be amusing to think beyond our little blue and green sphere and run the numbers for the rest of our planetary neighbors.

Interestingly, in the meantime there has been the explosive news that there very ----- well might be life on Venus¹https://www.nature.com/articles/d41586-020-02785-5–so what was supposed to be a mere fanciful exercise in number crunching may not be so far-fetched after all. Indeed, my handy-dandy guide to gambling at all your extraterrestrial social gatherings couldn’t be more timely!

Let’s get started, then, shall we? First, let’s review the ever so important equation²Figure credit: https://en.wikipedia.org/wiki/Birthday_problem that we use to calculate our probabilities, which we will subsequently plot and use to visually identify the 50-50 tipping point:

Note that we will be looking at the probability that there IS a pair of B-Twins, which is just the number above subtracted from 1.

Alright, inspecting this equation, surely y’all will recognize a rather familiar number, 365. To generalize this to other planetary bodies, we only need to substitute 365 with the number of days in a year for our particular planet of interest.

No problem! Let’s just Google those numbers:

For your viewing pleasure, allow me to plot those number of days:

Plugging those numbers into our magic formula in MATLAB, we quickly get this octant of plots:

The x-axes of those plots range from 30 all the way up to 600, perhaps making it tough to digest that information. Let’s plot just the Magic Numbers (@ p = 50%):

Oh, shit. Forget everything you know!

Did you catch that? That was a pretty cocky human move that I just pulled there–I took the lazy geocentric approach. I was measuring the length of the years in Earth days!

That makes no ----- sense, right? Why would a civilization on Jupiter measure anything in the length of the rotation of some planet they may or may not know about? We would never create calendars based on Jovian days!

What foolishness! Throw your old guide away!

Okay, so this is about where I fell down the Martian rabbit hole. Little did I know what I would be getting into when I started this little ill-advised adventure.

Apart from Earth and Mars, the question “how many days are there in a year?” gets weird pretty quickly. It was almost as deeply philosophical as the question I posed only days ago: “Can hair have hair?“

It makes most sense to measure a planet’s complete trip around the Sun (a “year”) in units of the time it takes to complete a full rotation on its axis (a “day”). I found a pretty informative astronomical resource³https://www.universetoday.com/37507/years-of-the-planets/ that helped me recalculate my “days per year” numbers.

Here are those revised numbers:

• Mercury: 0.5 dpy. Wait, whaaaat? A Hermian day is 2x longer than a Hermian year. Ok, so trying to work this into the framework of the Birthday Problem only made my head hurt. I tentatively promise that I will revisit the question of how they would theoretically construct a group of unique “Birthdays” at some later point in time.
• Venus: 1.92 dpy. Ugh. Just like with Mercury, I’m not going to even try to conceptualize Cytherean “Birthdays.” That’s going to be a whole ‘nother post on that topic. Also, here “day” is defined as sunup-to-sunup, which is not the same thing as a full rotation on its axis. You can start to see why things get messy, no?
• Earth: 365 dpy. Obviously there should be no change here. But we should note that we are ignoring Leap Days, etc.
• Mars: 668 dpy. Notice that this is slightly less than the 687 days measured when counting with the slightly shorter Earth day.
• Jupiter: 10,475 dpy. Shorter Jovian days result in over twice as many unique days per year!
• Saturn: 24,491 dpy. Like Jupiter, shorter Cronian days result in 2.4x more days on which some native Cronan could be born.
• Uranus: 42,718 dpy. A year on Uranus is like…no, wait, no time for juvenile puns.
• Neptune: 89,666 dpy. Uranus and Neptune both have shorter days, but not to the degree of Jupiter and Saturn.

Here is an updated visual graphic reflecting these numbers:

Alright, now we’re ready to crunch so numbers with some not-so-garbage input. Let me just lay out the 5 updated plots in larger detail for perusing at your leisure. If pressed for time, you can skip past them to the Summary Graphic.

I would like to just take a moment and point out something that I consider astounding. On Neptune, there are almost 90,000 unique days in a year–but you need only ~350 people before 2 of them share one as a birthday! That number is easily 10-20x lower than I would have ever intuitively guessed.

As you can see in our Summary Graphic (Figure 11), you’ll want to start attending larger parties if you really want to make some money off those poor souls born on planets past Mars.

Either way, you can consider yourself as prepared as ever to bum about about the solar system, hustlin’ your way into extraterrestrial infamy and fortune!

Now if you’ll excuse me, I need to get back to this very interesting article confirming what we’ve all long suspected: Uranus is the best.

What? Did you actually think that I would be able to escape the gravitational pull of a juvenile butt-pun?

Content created on: 10 Sept. & 13/14 Oct. 2020 (Thurs/Tues/Wed)

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