## Find a way to my Heart: The 1-in-10 Gay Problem Continued…

This post builds on the ideas from the first 1-in-10 Gay Problem post and takes a step into the creative world of **abstract maths**.

(!! This is the second in a series of three related posts – Part I “The 1-in-10 Gay Problem” & Part III “Hallucinogenic Maths”)

The following process demands your concentration but I have attempted to explain every step carefully. That said, if you struggle with anything please feel free to leave a comment and I will explain it in more detail for you. If it looks like there’s a lot of text here, scroll down to see what it all leads to – it’s well worth it – and it’s all necessary for the next post too, which will wow you for definite. So let’s make a start. But before we reiterate, consolidate, evaluate and titillate, we must get one thing straight…

**What is abstraction?**

Mathematical abstraction, defined brilliantly by Wikipedia^{[1]}, “is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.”

In English this means taking a real world system (like the gay problem), investigating the deep mathematical framework upon which it’s built and then analysing that in search of hidden patterns or bridges to other concepts. The intention of seeking these patterns is to develop a deeper understanding of the way in which systems work, which is itself the fundamental process in the development of both technology and philosophy.

**How can the Gay Problem be abstracted?**

The 1-in-10 gay problem is specifically based on the number 10. We have 10 people in a room, each with a 1-in-10 chance of being gay. In the first post we also considered the 1-in-2 problem, i.e. 2 people in a room, each with a 1-in-2 chance of being female. But we’re not confined to just 2 and 10; what about the 1-in-7 problem, or the 1-in-50 problem? *What about the 1-in-n problem?*

That’s right Fred. This is called generalising. This elusive “n” can represent any number you want it to. Describing a problem algebraically like this hugely increases the scope of an investigation and is almost always the first step in abstraction.

This is the 1-in-n problem: **“If the chance of somebody having a specified characteristic is 1-in- n, and there are n people in a room, what is the probability that there is at least 1 person in the room having that characteristic?” **(That characteristic could be anything, e.g. “having visited Italy”, or “will survive Swine Flu”, etc – it doesn’t matter.)

In the first post we established a formula for the probability of there being at least one gay person in the room:

**P(there is at least one gay person in the room)
= 1 – P(there are no gay people in the room)**

We can now generalise this formula to accommodate any characteristic instead of just whether or not someone is gay:

**P(there is at least one person in the room with some characteristic)
= 1 – P(nobody in the room has that characteristic)**

Concentrate here! To calculate the probability of nobody in the room having the characteristic, we have to consider each person in the room individually: we take the probability that an individual does not have the characteristic and multiply that value by itself *n* times (once for each person in the room).

This is exactly how we solved the gay problem; we took the probability that an individual does not have the characteristic, i.e. is straight – which was

9-in-10or0.9– and multiplied that figure by itself 10 times (n=10), i.e. 0.9^{10}.

In the general case, the probability that an individual does not have the characteristic is. The general formula then for the probability that there is at least one person in the room with a specified characteristic is:

Now for the interesting bit!

To help us to understand the behaviour of this function we can now construct a graph to show how it evolves as we increase the value of n:

But there is no reason just to consider positive values of n; **we’re in an abstract world now so we’re free to do what we like!** Here’s the same graph but for positive *and* negative values of n: (Note that the value of our function, i.e. the vertical axis, no longer represents probability because chance can’t be negative.)

What is significant here?

A good mathematician now tries to identify all the obvious characteristics of the graph, but we will concentrate only on the glaring jump between n=0 and n=1. The software I used to plot these graphs (Maple 12) is basically saying that the function cannot be evaluated between 0 and 1 [if you have a graphic calculator you can check this too – just plot *1-((x-1)÷x)^x*], or, in other words, that it doesn’t even make mathematical sense to consider the graph between 0 and 1. But is this strictly true?

Let’s take an example value of n between 0 and 1 and see what happens when we plug it into the formula: the obvious choice is n=1/2.

So our formula becomes which simplifies slightly to .

Raising something to the power 1/2 means taking its square root (see suggested reading), so it could be simplified further to . Taking the square root of a negative number is not allowed (it will be allowed in the next post!) so this is why our function “doesn’t work” between 0 and 1. **But I’m glad it’s not that simple.** Let’s consider n=1/3:

This time we’re ok, because **it is ok to take the cube root of a negative number** – there exists a negative number which can be “cubed” to reach -2 and that number is approx. -1.26. Similarly, if we consider n=2/3 we reach a “real” answer, approximately 1.59.

There are actually loads of feasible values of n; infinitely many to be “exact”! The values n=1/4 and n=3/4 behave in a similar way to n=1/2 – there are no “real” solutions – but if we consider n=1/5, n=2/5, n=3/5, and n=4/5 we get four more real number solutions. In fact, (jargon alert) we get real solutions for any value of n between 0 and 1 that may be written as a fraction with an odd denominator. (Capiche?!)

I wrote a program in Maple 12 (click here to view the program – I doubt it’s perfectly efficient but it does the job) to find a large number of these elusive solutions and here’s a version of Graph 2 with them all included:

And finally, if we remove the axes; focus on the domain 0<n<1 (i.e. values of n between 0 and 1); rotate the whole thing clockwise by 90° and then join the dots, we arrive at our hidden heart:

Are there any other hidden forms in this function? You bet! We’ll find a subset of them in the next post as we explore **an imaginary dimension!**

References:

- “Fractional Exponents” – www.mathsisfun.com – A guide to evaluating fractional powers

http://www.mathsisfun.com/algebra/exponent-fractional.html

**Acknowledgements:**

I wish to thank Martin Nelson for providing the bubblesort algorithm, without which my heart would have been broken, like this: (Thank you Martin; “thartin”)

Wow!

Incredible work Micky.

Good to see we’re not restricted to integer numbers of people or even integers.

My question to you …

Are there countably infinite values of n which give ‘real’ solutions or an uncountable amount?

And if there are, can you show a bijection that maps the natural numbers, (funny double lined N) onto the set of numbers that provide real solutions.