216: Romantic Drama Equation
|Romantic Drama Equation|
Title text: Real-life prospective-pairing curves over things like age can get depressing.
| This explanation may be incomplete or incorrect: Even since Randall himself is stuck by wolfram-alpha, a better solution has to be done here.|
If you can address this issue, please edit the page! Thanks.
The equations in the comic and the graph show how many different love pairs can be made if you know the number of females and males in a group. The text explains that it was inspired by TV Romantic Drama (in this case, the gay drama Queer as Folk), but of course the formula is valid for any group of people. There are two graphs and equations - gay option is the case when we are looking for pairs with same gender, straight option in for heterosexual equations. The interesting/funny part about the results is that in most cases there are more possibilities when we consider the homosexual option. Also it is interesting to observe what is kind of obvious - in the heterosexual case the "best" case is if both genders are present equally and the possibilities drop very fast if there is substantial difference between genders.
It should be noted that the chart assumes that the ENTIRE cast, male AND female, will ALL be of the same sexuality (homo OR hetero).
The graph makes a note that it only holds true for large casts. Case in point, with a cast of only four people: a two-to-two female-to-male ratio will have four straight pairings to two gay pairings, while a three-to-one female-to-male ratio will have three straight pairings and three gay pairings.
The title-text mentions that Randall made a chart of his own prospective dating pool as he gets older, and was depressed by the results.
The formulas may be derived as follows:
Each straight couple needs to include one of the x males and one of the (n-x) females so there are x(n-x) possible ways of combining one of each. E.g., if there are n=5 people, of whom x=2 are male, then there will be 3 possible pairings involving the first male, and three possible pairings involving the second yielding 2(5-2)=6 possible pairings.
Each gay couple needs to include either two males or two females. To choose two males, we can start with any of the x males and choose any of the (x-1) remaining males. However, that counts each possible pairing twice. E.g., Adam&Steve got counted when we chose Adam first and Steve second, and again when we chose Steve first and Adam second. To avoid double counting the possible couples, we therefore need to divide that total by 2. So there are x(x-1)/2 possible male-male pairings. Similar reasoning involving the (n-x) females tells us that there are (n-x)(n-x-1)/2 possible female-female pairings. Multiplying these out and combining the male and lesbian couples together, we get the total number of possible gay couples is [x^2 - x + n^2 - nx - n - xn + x^2 + x]/2. That simplifies to [n^2 - n + 2 x^2 - 2 xn]/2. The left two terms can be combined together as n(n-1) and the right two terms can be combined together as 2x(x-n). Since the sum of these terms was divided by 2, we get that the total number of possible same-sex pairs is n(n-1)/2 + x(x-n), which is what the cartoon says.
Equations and links to wolfram-alpha:
- Formula for Gay pairing: n*(n-1)/2+x*(x-n)
- Formula for Straight pairing: x*(x-n)
- TV Romantic Drama Equation (Derived during a series of "Queer as Folk" episodes)
- [A table shows equations for possible romantic pairings in a TV show. The equation under "gay" is n(n-1)/2+x(x-n); the equation under "straight" is x(n-x).]
- x: Number of male (or female) cast members.
- n: total number of cast members.
- [A graph plots pairings (for large casts) against cast makeup. Each of the above equations forms a curve. "Gay cast" starts high for an all male cast, dips down at 50/50 cast makeup, and then rises again for all female. "Straight cast" starts at zero for an all male cast, peaks at 50/50 cast makeup, and then drops to zero again for an all female cast. The two curves intersect at two points close to the middle.]