This essay is Public Domain. You can freely copy it.
August 2003, Henrik Ingo
Over drinks this guy tells me the story of how he met his fiancee. It was a truly remarkable story. He is Finnish, She is Portuguese, they meet in England, in an "astonishing woman just walks into your life" way. But the incredible story is, at meeting-the-parents time, they find out that their parents have once met and He actually has been to Her home in Portugal before she was born.
A woman that's also listening to the story bursts out: "Incredible, what are the chances of that! It's like winning on the lottery!" After a while they notice that I'm slightly absent. "What's with you?".
I reply: "Interesting question, but I would need a calculator to find out..."
So I later write him this email...
Let's start with the easy part: What is the probability of winning the Finnish Lotto-lottery? This question is certainly a common excercise in any probability class. Sadly, this is not a guarantuee that I will get anything of this right, since I never really got on top of this part of my studies. Kind of an alarming fact really, given that tomorrow I will start working on new coursematerial in statistics for my university. Luckily I will only be responsible for publishing web material, not actually writing the math.
In the Finnish lottery, they draw 7 numbers out of a total of 39. There are thus
39 * 38 * 37 * 36 * 35 * 34 * 33 = 77 519 922 480
possible combinations, or I shouldn't say combinations, because in math the above is called a permutation. A permutation is something where the order of the numbers (or whatever items) is important. But in Lotto the order in which the numbers are drawn is not significant. 1,2,3,4,5,6,7 and 7,6,5,4,3,2,1 are the same as far as Lotto is concerned. It turns out that there are:
7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040
ways to draw the same set of 7 winning numbers. Therefore the amount of different sets (called combinations, confusing isn't it) is only:
77 519 922 480 / 5040 = 15 380 937
If you buy one ticket the chance of winning is the inverse of that
1 / 15380937 = 0.0000000650155
or, expressed as a percentage
0.00000650155 %.
In other words, the change of winning the lottery is about 1 in 15 million.
We are now ready for the more interesting part of our story. What is the chance of Him and Her meeting in the way they did?
Their story seems to be one of many incredible coincidences. A true mathematician would probably not even embark on a task of calculating the probability of it happening. For instance, what is the chance that an uncle that has already been on Inter-rail twice, will still be interested in doing it a third time? And if he does it, and ends up in Paris, what are the odds that he will then head for Portugal altough he was not planning on doing that, and there get to know Her parents? Breaking all this down into smaller and smaller pieces, is like measuring the length of the shoreline on a beach. The length increases indefinately when you use more precise tools of measurement. (This is because measuring the path around a stone is always longer than cutting right through it. So as your precision increases you will be entangled zigzagging around every little stone, then around all the small particles of sand, and eventually, every atom that constitute the little sand particles, thus stretching the path indefinately.)
But you're in luck, because I'm not a mathematician. So let's just do what we always do in physics: Ignore the difficult parts and if they can't be ignored just make a good estimate. The estimate most often being just an educated guess, perhaps even an uneducated one.
So let's assume that we have Him living in Finland, and Her living in Portugal and their families have the history they have, wich we for simplicitys sake will not discuss any further. What are the chances of Him and Her meeting?
According to the US Census bureau (http://www.census.gov/ipc/www/idbnew.html) there are 535 394 880 men and 508 593 872 women between the ages 20 and 29 in the whole world. It is true that She being the younger might very well have met a 19 year old boy, or He could have met someone over 30, but we'll take these numbers as an estimate for the size of a random ten year range of young people of the opposite sex.
It is difficult to estimate the amount of people that you meet during your life or youth, so let's narrow it down to the people you date during your life. Having only dated my wife since after high school I might be the wrong person to make even that estimate, but it seems reasonable to state that most young persons will date between 0 to 30 representants of the opposite sex, with a small group of people heading for figures higher, some significantly higher, than that. From this argumentation it seems that 10 is a good round number for the number of people you ever date.
We can now state, that the probability that two persons somewhere out there meet and start dating, is approximately
10 / 500 000 000 = 0.000000019662
or 1 out of 50 billion. Winning the lottery is a piece of cake compared to this! (Remember, in Finnish 1 billion is "1 miljardia" not "1 biljoonaa".)
Of course this was only a very average calculation. To get a more exact result, we should weigh the probability of dating any one person with the geographical distance between the two. After all, it is not that incredible to end up with someone that grew up in the same town as you did. But it is much more improbable to meet someone that lives at the opposite edge of your continent. Further, it is almost impossible (but just almost) that you ever end up with a girl that lives in the Sub-Saharan rainforest.
Again, we will obviously not get into a detailed calculation like that, but let's look into that direction, making yet another simplification, to get some perspective into the problem at hand. Let's narrow the problem down to Europe-only. In effect, we are rounding the probability of ending up with a non European down to 0. Let's then say that you are 1000 times more likely to end up with someone from your own country, than someone from another European country. (This is equivalent to saying that 3000 Finns are married with a foreigner. Perhaps a slight under estimate.)
Putting in geographical weights also makes the calculations assymetrical, that is, we would need to do separate calculations for Finland and Portugal. So let's focus on the probability of Him meeting Her.
According to the US Census there are
316 683 Finnish women between ages 20-29 50 295 392 European women between ages 20-29
We then use the derived formula
[prob. of meeting a European [people to date] woman that is not Finnish] * ---------------------------------- [European women] - [Finnish women]
= 1/(1000 + 1) * 10 / (50 295 392 - 316 683)
= 0.000 999 000 999 * 10 / 49 978 709
= 0.000 000 000 199 89
or 1 in 5 billion. Slightly better odds than last time, but still in the same rough order of magnitude. Taking into account all the Finns that end up meeting Americans and Australians, and other non-European women, would again make the probability smaller and we would probably come very close to the initial estimate of 1 in 50 billion.
Of course we have not at all discussed the probabilities of all the incredible events that led to their families getting to know each other, nor the chances of their story to then develop in the way it did and them meeting how they did. All these incredible events would yield an infinately smaller probability, just like measuring the length of the shoreline. However this short calculation is enough to establish proof, that the next time He tells this story to his barmates, and someone says: "That is incredible, like winning on Lotto", you can reply: "You know, winning on Lotto is nothing ..."
For the more mathematically inclined.
Altough I wrote this completely with my tongue in my cheek, reflecting back on it I think that the calculations - average they be - actually make a lot of sense. One could argue, that the above calculations hold, if you assume that:
quod erat demonstrandum.
This essay is Public
Domain. You can freely copy it.
August 2003, Henrik Ingo