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The puzzle of the two Black Queens

What number does your intuition tell you to pick?

Many mathematicians, including me, are fascinated by puzzles that have counterintuitive answers, and I recently came across a lovely example that I haven't seen before.*

Imagine, as so often seems to happen in puzzle world, that you have been captured by an evil villain who has threatened to keep you locked away for the rest of your life, unless you happen to strike lucky.

He takes out a regular pack of cards and gives them a thorough shuffle.  "Now," he declares, "I am about to turn over all 52 cards, one at a time, but first I want you to predict the position in the pack of the second black queen that I turn over.  If your guess is correct, I will release you."

Your odds are not high - there are 52 cards.  Two of them are black queens - the Queen of Clubs and the Queen of Spades.  And only one of these will be the second black queen to be turned over.  But if you want to make your chances as high as possible, which position in the pack should you choose?  Hint: don't pick the top card (number one), because by definition that can't be the second black queen - though it might be the first.

Before you read on, pick a number in the range 2 to 52 where you think the second black queen is most likely to be.

Spoiler alert - the solution is coming.

Most of us have some sort of intuition for problems of this kind.  Perhaps the most common intuitive reaction is to think as follows: "Since the second black queen could be in any position from 2 to 52, and its position is completely random because the cards have been shuffled, I may as well pick any number I like between 2 and 52, it makes no difference."

With similar reasoning, some people go for the middle, number 26 or 27. (Which one is the middle card? See my previous blog about Robin the singing muppet.)

A bit more thought might get you to adjust your answer.  The only way your card could be in position 2 is if the first black queen is at position 1, and that's pretty unlikely.  Whereas if you choose, say, position 40, there are lots of places the first queen could be.  So maybe your card is more likely to be in the second half of the pack than in the first half, three quarters of the way down for example.

In fact there is one position that is more likely than any other.  The most likely position of the second black queen is position 52, the bottom of the pack.

How can this be?

One way to see it is to work out where the FIRST black queen is most likely to be.  The chance that the top card will be a black queen is 2/52 (= 1/26), because the top card could be either of the two black queens.  The chance that the second card is the first black queen is a bit harder to work out.  It's the chance that the first card ISN'T a black queen (=50/52) multiplied by the chance that the second card IS (=2/51). The chance that the second card is the first black queen is therefore 50/52 x 2/51, which is slightly smaller than 1/26. And in fact as you go through the pack, the chance of the next card being the first black queen keeps reducing until it is zero when you get to card fifty-two.  In other words, the most likely position of the first black queen in the pack is that it will be the top card - a chance of 1/26.

Now imagine turning the pack of cards upside down.  The first black queen you encounter is the same as the second black queen when the pack is the right way up. (Think about it).  So by symmetry, we have exactly the same calculation as before when we were working out the chance of the first black queen.  Hence the most likely position of the second black queen is that it'll be at the bottom of the pack, in position 52.  

Let's not get too excited, the chance of it being there is only 3.8%, so your odds still aren't great.  But when it comes to escaping from evil villains, every little helps.


* Not everyone will think the answer is counterintuitive, but remember that intuition is the answer that comes to you without you doing any serious thinking about the problem.