What are the odds that there is no solution on a given Set board?

I was playing a game of Set with a group. We had the usual twelve cards out and all agreed that there was no set. We then increased the number of cards to fifteen and were still unable to find a set. (Unfortunately I no longer have a picture of the board in question, so it's impossible to verify.)

Is this even possible? Were we all just really, really tired? What are the odds of this occurring?

• I know for a fact I've had it happen to me, with 18 cards. I don't know the exact odds, but I do know it's possible. – Xynariz May 30 '14 at 15:21

Wikipedia says the chance of no set being present in 15 cards when playing is $88:1$. This assumes that the first twelve cards have no set. It also says you can find twenty cards with no set.

I'm not sure about the probabilities, but I thought I'd address the question "Is this even possible?".

Here's a favourite paper of mine, The Card Game Set. In it they give a collection of twenty cards with no set, which is apparently the maximum (i.e. 21 cards must contain a set somewhere):

So yes, it's perfectly possible to have no set within 15 or even 18 cards!

• That set board is so infuriating. There are way too many almost-sets. – Joe Z. Sep 11 '15 at 15:07