by Burkard Polster and Marty Ross
The Age, 8 June 2009
Much to our surprise, we were recently involved with a fully functioning Southern Star. No, not the giant wheel at Dockland that remains immobile. Copyright 2004-∞
In fact, we were contacted by Matthew Dunn, a reporter for The Great Southern Star, Leongatha’s local newspaper. Matt wanted to have a chat about a rather amazing coincidence. It seems that the Chapman family of Yarrawonga had just celebrated a new arrival, a bouncing baby Ebony. Ebony was born on May 6, the birthday of her great grandfather, Mark Chapman.
That’s notable, but hardly enough to make the news, even in Leongatha. However, Ebony’s sister Brodie also shares a birthday, October 28, with their grandfather, Ross Chapman. Moreover, their older sister Abbey shares her birthday, April 14, with their father, Mark Chapman. Now that’s pretty cool!
Matt contacted us, hoping to discover exactly the chances of all this happening. We obliged as best we could, but it was not as straight forward as one might imagine. The tricky question is, what exactly has happened?
At the simplest level, the chances of two people having the same birthday is about 1 in 365. So, for three pairs of matched birthdays we multiply 365 x 365 x 365, giving about 1 chance in 48 million.
Perhaps we also want to include the fact that it is three sisters matching with the Chapman men. There is 1 chance in 8 that the three children will all be female, and so the overall chance is out to about 1 in 400 million.
However, perhaps the odds are not so extreme. Imagine, for instance, that the three children had matched birthdays with the men, but in a different order. Undoubtedly, Matt would have still contacted us to find out the odds of this “happening”. But, with 6 ways to match children to adults, the odds are down to about 1 in 8 million.
We can go further. Consider all the parents and grandparents and great grandparents together, 14 in all. What are the chances of the three children matching with any three of the adults? There are 364 ways to choose the three adults, and the overall odds are then about 1 in 130 000. Still impressive, but not in the millions.
So we have hugely divergent answers: roughly the difference between winning Division 1 or Division 2 in Tattslotto. Which answer is correct? The answer is, there is no answer.
Probability is about the analysis of repeated “happenings”. In order to calculate the probabilities, we must declare the precise events that might occur. With a coin or a pair of dice that is easy, but in real life it is not so obvious.
Before an event has occurred, we are pretty much forced to declare the exact event we are imagining. However, after the fact, we can characterise that event in all sorts of ways. For each such characterisation, we’ll have a different estimate of the probability, each with equal claim to correctness. Further, our innate desire to see patterns means we’re very likely to see, or imagine, the specialness of events. This results in estimating probabilities at the high end.
Think you’ve got it? Then perhaps you can solve Matt’s follow-up puzzler. After his report on the Chapmans, Matt received an email from Sue Cashin. Sue and her two sisters all have the same birthday. Sue also has a granddaughter with the same birthday. Sue also has five brothers, but with different birthdays. So, what are the chances? We’ll entertain all suggestions between 1 in 100 and 1 in a million. Our own thoughts will appear on our website on the weekend.
Probability theorists know well the danger of ill-defined events, and the distorted view of hindsight. Alas, some others do not. Consider the current fad of Intelligent Design, religious creationism but with the gloss of mathematical respectability. ID-ers spend much time examining bacteria and whatnot, arguing the mathematical unlikelihood of certain biological characteristics. Unfortunately, the same uncertainties and flaws in our Chapman calculations are rife in these ID estimates. Einstein famously declared that God doesn’t play dice with the Universe. The flipside is: dice are a very poor way to discover God.
www.qedcat.com
All rights reserved.