Oh, man. You’re going to be so ----- rich.
My mom has 11 grandchildren. Of those, the first two cousins missed being born exactly one year apart by less than one day.
Over two decades later, the Boss Lady went into labor with our first daughter (aka “the Elder”) late in the evening two days before the first of these two birthdays. Making the obvious assumption that the Elder would subsequently be delivered within 24 hours, I was all nerding out over the fact that Mom would have grandchildren with birthdays on 3 consecutive days. You know, like some sort of Grandkid Birthday Bingo or what-not.
Well, that rascal took 36 hours to show up, so instead of getting a Bingo, Mom finally got that pair of Birthday Twins she had just missed out on 21 years prior. Oh, right…what are Birthday Twins, you ask? Well, they’re simple two non-twins who share the same birthday.1Typically the birth year is ignored.
I suspect Birthday Twins, like phantom flatulence, must run in our family.2Two of my sisters are for-realz twins, FWIW. When I was in high school in Podunkville, Kansas, there were somewhere between 15-17 of us in my entire class. Yet, somehow within that small group, I was privileged to be a B-Twin myself. Even better, we not only shared the same birth-year (obviously), but we even had the same first name, LOL. What are the odds?!?
Seriously, though: what are the odds?
For simplicity’s sake, we’ll stick with the basic case of celebrating your birthday the same day of the year as someone else. Examining the odds of having the same name and birth-year, as in my case, is, as the academics say, “Beyond the scope of this text.”
I was introduced to this so-called “Birthday Problem”3https://en.wikipedia.org/wiki/Birthday_problem on the first day of my philosophy class in college, and have been infatuated with it every since. This is actually one of the more well-known examples of how, by default, we humans are pretty dang terrible at estimating probabilities and assessing risks. And it all stems from fairly innocent-looking question: “For a group with a certain number of people in it, what are the odds that two people share the same birthday?”
But I say we should, as a shady character might say “make things interesting”–let’s put some hypothetical money on the line.
Say you’re a well-travelled social butterfly with a gambling addiction, living in the times before COVID-19. You attend many parties and gatherings all throughout the year, of all different sizes.
Feeling the irresistible urge to feed your habit, you need to find a sure-fire way to make bets without ending up in the Poor House. Enter the Birthday Problem–which, by the way, is actually a pretty great icebreaker at social gatherings in real life.
You know that since you go to so many parties, you can always put money on two people having the same birthday any time the probability of that being true is at least 50% or above, and bet against it otherwise. Over many parties and many wagers you’re statistically guaranteed to come out on top.
As long as you know roughly how many people are at the party, you can be confident whether those odds are above 50% or not.
Actually, the question that you need to answer is simpler than that: “How many party peoples need there be present to have at least a 50% chance of finding a pair of B-Twins amongst them?”
That’s right: one number. You don’t to memorize any fancy formulas and calculate them in your head in real time. You only need to know one number to guide your foolproof betting scheme.
Oh, man. You’re going to be so ----- rich.
Now, let’s run the numbers…
Bear with me, as I’m doing this from memory, instead of being smart and just googling it. FYI, my probabilities run between 0 (ain’t happening, ever) to 1 (it’s a certified irrefutable fact), which translates to 0%-100% in everyday-speak.
The key to this is asking the right questions. The first questions is: what are the odds 2 people don’t have the same birthday? Then you only need to subtract that number from 1 to get the probability that they do. So:
Pyep(pp) = 1 - Pnope(pp),
where Pyep is the probability that “yep, we got a pair of B-Twins up in heeeer,” while Pnope is the probability that “nope, they all be a bunch of unique snow-flakes in these parts,” and both of these are functions of pp, the number of Party Peoples present.
This one is pretty simple. The first person can lay claim to 1 out of the 365 days in a regular year, leaving 364 days that the other person can have without them making a pair.
So we have:
Pnope = (364/365) ~ 0.99726 (99.726 %), for pp = 2
This puts Pyep at 0.274%–roughly a quarter of 1%.
Now bring in another person. Two days of the year have already been claimed, leaving 363 days for the third person. This probability needs to be multiplied by the probability that the first two people didn’t have the same birthday:
Pnope = (364/365)*(363/365) ~ 0.9918 (99.18 %), for pp = 3
At this point Pyep almost quadruples to 0.82%, so we can see that this isn’t linear. Why is it important that it is not linear? Because linear usually == intuition. Intuitively, humans are pretty good at linear extrapolation: “Oh, housing prices have gone up steadily over the last 5 years; no doubt that will go up by the same amount over the next 5 years!” (Note: this intuition would typically be wrong; see 2008.)
And so it goes: with each new person, the number on top decreases by 1 day, and that fraction is multiplied by the previous Pnope. At this point I’m going to cheat and use a screenshot from Wikipedia4https://en.wikipedia.org/wiki/Birthday_problem to show you the general equation:
It may look kinda scary, but don’t worry: we don’t have to do this by hand or in our head. It’s not that bad if you have a good calculator or math software. To that point, I took the liberty of plotting it for you in MATLAB:
TWENTY-THREE PARTY PEOPLES. Not only is that going to be the name of my next band, but it is also the answer to your poverty problems. Twenty-two party peoples or less? Bet against Birthday Twins. Twenty-three party peoples or more? Bet on there being at least 1 pair in the crowd.
That’s all you need to know!
The point of the story is that if you can accept that your intuition might not always be right–and you know how to ask the right questions–you’re going to be rich.
Oh, man, you’re going to be so ----- rich.
Content created on: 10/25 September 2020 (Thurs/Fri)
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