What are the odds?

A friend recently commented on Facebook:
My partner and my sister have remarkably similar taste in birthday cards!! One (card!) was bought in Whyalla, the other in Coffs Harbour. What are the odds?!
He helpfully included an image of the two identical cards.
In various works by Sir Terry Pratchett, set in the Discworld Universe,  he mentions that the problem with million to one shots is that they always happen.  One book - sorry, can't find the exact reference - a bunch of wizards are trying to set something up so that the odds of it happening are exactly 1,000,000:1.  They keep getting close (eg 999,995:1 or 1,000,005:1) so say things like "Put it in a purple sweater and orange tie."  Eventually they get the million to one ratio and the problem is solved

So my first guess is the odds are 1,000,000 to 1.

I then tried some slightly more rational approaches.

The first of these started by looking through Google, for spots in Whyalla that sold birthday cards.  I came up with Wild Cards, which said they were a franchise, so I thought I was on a winner.  Apparently this is part of the W H Smith empire which has stores all over the country.  Unfortunately not in Coffs Harbour.  A similar search for Coffs Harbour was fruitless.  So that approach fizzled out.

My next trick was to search for Bilby Birthday cards.  The best bet here was a card produced by Koala Hill Crafts, (KHC) which featured 4 Bilbys in very very similar poses to those shown above.    The design looks slightly different but I am almost certain that's the one.  The designs are protected (fair enough as well) so I can't get a good copy but this gets across the idea.
They are a wholesale operation as well as online retail so, given the professionalism of their anti-download software, I'd take a fair guess that they have marketed their products fairly well.  This should include a national chain such as the Wild Cards (which positions the cards in Whyalla) and major tourist haunts (which gets the cards to Coffs Harbour, although I can't link card sales to the Big Banana).

Now the penultimate step in calculating the odds is that the Bilby cards are branded by KHC as part of a Whimsies series.  Given that the recipient of the cards produced, for many years a series of articles under the headline of "AvIan Whimsy" someone looking for a card for him would spot that brand and home in like a magnet to an RSJ.  (I had some other metaphors involving Miura bulls/red capes and Tstetse flies/blue shirts but decided that, while the natural history linkage was sound, they were not appropriate for the ladies involved.)

Finally there are 37 designs in the Whimsy series.   So given that the donors were lurking in locales where the brand was evident at the time they needed a card the chance they would go for the Whimsy series is about 100%.  A second order probability is 1 in 37 that either donor, if they chose at random, would go for the Bilbies and I think this means that for both the independent events to arrive at the same outcome would be 372 or 1 in 1,369.  However:
  • I thought the design of the Bilby card was the best;
  • Bilbies are cute (which, dancing amongst a minefield of accusations of sexism, could be very attractive to sisters and blokes partners): and 
  • Bilbies are at risk so would be an item close to the card-recipients heart.
This leads me to conclude that Sportsbet, if they were framing a market on this proposition would offer a price around 5:1.


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