Posted by: caitlincrump | November 17, 2008

Ace of Spades or 2 of Clubs?

OK… So I heard some people discussing this problem awhile ago, and I wrote a note about it because I thought it would be a kind of fun blog post.  Here goes…

So you have a deck of cards and you shuffle them so that they are completely random.  Then you turn them face up one at a time until the first ace appears.  Which is more likely, that the next card be the ace of spades or the 2 of clubs?

It helps to think about this problem in terms of the cards in a row.  So, if you had all of the cards  lined up in a particular order so that the ace of spades follows the first ace, then there are 51other cards to arrange and there are 51! ways to arrange them.  Since there are 52! ways to arrange all of the cards in the deck, the total probability is

51! =   1

52!       52

Now, we could use the same logic to arrange the cards with the first ace before the 2 of clubs, which would give us the same probability.

This means there an equal chance of either of these two cards appearing after the first ace.  It seems intuitive once you realize that there is an equal chance of ANY card appearing after the first ace.

I know this problem may seem a little simple or silly, but it really surprised me how much these people were struggling with it.  They kept claiming that the ace of spades had to have a less likely probability, because the first ace could be the ace of spades.  However, then someone said, what if the 2 of clubs appears before the ace of spades?

I guess it just depends on how the problem is phrased more often than you would think.  I can’t tell you how many times I’ve felt totally confused by a Math 152 problem, and then realized that it was the phrasing of the problem that had tripped me up.

That’s about it from this end.

Sorry, no witty jokes… just go down a few postings for those.



  1. Great! It’s wonderful when a seemingly complicated problem becomes simple from the right angle.

    This is a good example of one of my little habits: I like to take a sort of `kinesthetic’ approach to probability. In this case, I imagine myself removing the ace of spades or 2 of clubs from the deck, then arranging the 51 other cards in a row. I know that there are 51! ways of doing this. Then I simply insert the card I hold, whatever it is, right after the first ace in my row. I know I can do this because there’s at least one ace among the 51 in the row already, and there’s exactly one which is the “first”. By imagining the sequence of events, it makes the probability seem very intuitive.

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