Sunday, January 26, 2014

Frog kissing. Or, why do we never learn?

A friend pointed me to this story about the average number of men (or rather frogs) a woman has to date before finding their 'Prince Charming' (apparently it's 15).  Let's leave aside for now how they came up with the number: I've no doubt the methodology conforms to the most rigorous standards that we've all come to expect from the internet.  I thought that this paragraph might give some people hope:
If you've made out with 11 men, great news! A new survey suggests you're only 4 away from Prince Charming. On average, anyway.
It's now my duty to crush that hope.
 Distinguishing princes and frogs can be difficult at first glance [Photo: Allen Warren]

The bad news is that (even if the rest of it is correct) the survey absolutely doesn't say that.  If you've been in 11 disastrous (but possibly ultimately affirming) relationships, you've guaranteed that the total number you end up in is at least 11.  But it could have been fewer than that (it happens apparently), so the possibility that you stayed with your childhood sweetheart is included in the average which arrives at 15.  Once you exclude that possibility, the average will only increase.

It's the same with life expectancy: every year you live, your total life expectancy will increase simply because you didn't die that year (way to go!).  When you're young and healthy your chances of dying this year aren't high, so it's no great surprise when you survive; your life expectancy therefore remains roughly the same.

If you're 25, male, and live in England and Wales, your remaining life expectancy is 54.71 years, as estimated by the ONS (data from 2010-12), for a total of 79.71 glorious years.  If you're 26, your remaining life expectancy is 53.74, giving a slightly increased total of 79.94.  This becomes much more pronounced for older people: if you're 85 and female, you have a remaining life expectancy of 6.84 years, for a total of 91.84, but by the time you're 86, this has increased to 92.36, a gain of more than half a year, and 10 years more than at birth.

In fact, it's quite possible for the remaining life expectancy to increase, let alone the total.  Imagine a population in which half the people die as infants (under 5), but the remainder live to be 75.  When born, your life expectancy is about $\frac{1}{2} \times 5 + \frac{1}{2} \times 75 = 40$: not great.  But if you make it to age 5, you're guaranteed to live another 70 years.  Sadly, high infant mortality means that in some parts of the world this is not an abstract concept.

Kermit's Revenge

One particularly interesting case is if the distribution is memoryless.  Let's get back to the dating: suppose your approach is to just meet people at random, learn nothing from each experience, and you don't become more or less picky over time.  So we assume that, each time, you have a fixed probability $p = \frac{1}{15}$ of ending up in a relationship that works.

The chance that it takes exactly $k$ relationships before you find a non-amphibian is $(1-p)^{k-1} p$, because the first $k-1$ have to all be unsuccessful, and the $k$th one a charm.  If you can sum a geometric series (hopefully my first year students are reading this), then you can show that the total number of people you have to meet before settling down is, on average, $\frac{1}{p} = 15$.

But if you've already dated 11 people?  Well, we can use conditional probability: the chance of you having to date exactly $11+k$ frogs and princes is
$$(1-p)^{11+k-1} p,$$
and the chance that you find 11 people without meeting a keeper is $(1-p)^{11}$.  So the conditional probability of having to date exactly $k$ more guys, given that you've already put up with 11 toads, is
$$\frac{P(X = k+11)}{P(X > 11)} = \frac{(1-p)^{11+k-1} p}{(1-p)^{11}} = (1-p)^{k-1} p.$$
But this is just the chance of having to meet $k$ guys in the first place!  This is the memoryless property.  It doesn't matter how many times you've tried and failed, you're still at square one until the prince comes along.

Of course, we might like to think that we learn from our mistakes, grow as people, and make better decisions as we get older, but the empirical evidence available to me doesn't bear this out.