As some of you may recall, I kicked off the Fun with Excel series with a post on attraction, where I hoped to explore the mechanics of physical attraction from a statistical perspective. Due to the amount of feedback I have received (positive, skeptical, or otherwise), I have decided to write a follow-up post.

In the first part of The Laws of Attraction, I focused solely on physical attraction, and the impact of bias in our perception of attractiveness on seeking a compatible partner. In part two, I focus on the bigger picture: given a set of personal traits, what is the probability that you will find someone with those traits at the specific level that you desire?

Background: In the song One In A Million, Ne-Yo sings about a girl who he calls “one in a million.” Of course, not content with just enjoying the music, I wondered to myself what it actually meant to be “one in a million.” One way of measuring this is by breaking down attraction into a larger set of personality traits and trying quantify our desires, which is essentially what online dating services do with their “matching formulas.” For purposes of our exercise, let’s say you have a list of 10 distinct characteristics that you believe to be important and that you actively look for when searching for a partner. You might be more picky on some traits than others, but it isn’t too hard to quantify your objectives. Similar to my previous project, I quantify these objectives in terms of percentile, which, at least from a guy’s perspective, is pretty straightforward. For example, I might say, “I’m only interested in a girl who’s in the 80% percentile for Trait 1, 90th percentile for Trait 2, 50th percentile for Trait 3…” and so on and so forth. Now, the question is “what are the chances that such a girl exists?” A closely related question is “how many such girls are out there?”, followed by the not-so-fun reality-check of “what are the odds that I’ll actually find such a girl?”

The Model: While we won’t tackle the last question in this post, the first two are pretty straightforward to simulate from a mathematical standpoint. For each trait, the probability of finding someone who is at the X-percentile or higher of that trait is (100-X)%. For multiple traits, all we have to do is multiply these probabilities together, but the key assumption here is that all the traits are independent. Obviously, this isn’t true in real life, but we’ll revisit this point in a little while.

Assuming we start with a set of 10 traits, I will define a person having N “Perfect” Traits as someone who ranks at the 90th percentile or higher in N traits, and at the 50th percentile or higher in the remaining 10 minus N traits. Thus, assuming a world population of 7.12 billion, a male/female split of 50/50, and that you are heterosexual, the number of potential partners with 0 “Perfect” Traits  walking on the planet is 3,476,563, or 1 in 1,024 (the mathematically inclined should immediately realize that 1,024 = 2^10). On the other hand of the spectrum, there are theoretically only 18 people with 9 “Perfect” Traits, or 1 in 200 million. Note that a person with 10 “Perfect” Traits technically doesn’t exist, as probability indicates a 1 in 10 billion chance. At this point, the astute reader will note one possible answer to Ne-Yo’s earlier problem: if you consider a smaller set of 6 traits rather than 10, a “one in a million” girl would simply be a girl who has 6 “Perfect” Traits (all 6 traits at the 90th percentile or higher) in that scenario.

The Results: I plotted the entire spectrum of N “Perfect” Traits in the scenario of 10 traits, to arrive at the following graph:

It should be no surprise that our graph strongly resembles a normal curve, as we are working with a binomial distribution.

I suppose the lesson here is that it doesn’t pay to be picky, but recall the very important (and incorrect) assumption we made earlier that all traits occur independently of one another. In the real world, however, this couldn’t be further from the truth. Creativity may be correlated with Curiosity, Honor may be correlated with Kindness, and Intelligence may be highly correlated with (or the cause of) all the other traits. Accounting for the dependencies between and among all 10 traits would require us to estimate both marginal and conditional probabilities, which would not only be difficult, but also complicate our model very quickly. Statistical mumbo jumbo aside, what this means is that the probabilities estimated by a simple binomial model are far too conservative (too low). This should be great news for all the picky daters out there.

An alternative way of tackling the dependent traits problem is to simply consider a smaller set of traits. For example, if we created a list of 10 traits, and then realized that two of them were very highly correlated with each other, then we could eliminate one of them and simply consider a 9 trait model, which in turn would be a more accurate simulation of what the actual probabilities might look like in real life. To that point, I also plotted out graphs for scenarios involving 7 traits and 5 traits:

Note that as we decrease the number of traits, the number of potential candidates increases exponentially. So if you only considered 5 main traits, and furthermore were only picky about 3 of them (3 “Perfect” Traits in the graph above), then you would only be looking at a probability of  1 in 400. Not bad.

Conclusion

At the end of the day, it is perhaps silly to attempt to model real life human dynamics with 50 lines in Excel. But that would also be missing the point of the exercise. Thinking about real world problems from a different perspective (whether it is psychologically, statistically, or otherwise) can shed new light on the issue, or simply affirm something we already knew or suspected. Even if it is only the latter, there is still value derived from being able to connect the dots between a variety of different frameworks.

As for me, my dream girl in the 10 Trait model is about 1 in 5,925,926, and about 1 in 53,333 in the 5 Trait model. I’m not sure if I’ll ever find her, but it’s satisfying to know that she’s out there.

-J