Probability Problem: Queen of Spades?

[Blue_Jaunte]

Homework Statement

A friend lays out three playing cards on a table, from a pack of 52. He tells you that one is definitely a queen and the other two are definitely spades. What is the probability that the middle card is the queen of spades?

Homework Equations

The Attempt at a Solution

The probability that the queen is the queen of spades is 1/4. The probability that either of the spades is the queen of spades is 1/13. So the probability that any of the three cards is the queen of spades is...?

This isn't a homework problem. I just made it up to help me fall asleep, but now I can't figure it out.

 

[hgfalling]

Well, let's start with combinations.

There are 52C3 = 22100 three card combinations.
How many don't contain a queen? 22100 - (48C3) = 4804.
Of the ones that contain a queen, sort them by highest ranking queen:

4 ways there are three queens: Qs 3, Qh 1
6*48=288 ways there are two queens: 144 Qs, 96 Qh, 48 Qd
4*24*47 = 4512 ways there is one queen: 1128 for each queen.

When there are three queens, conditions are never met.
Where there are two queens, conditions are met IFF Qs is present with one other spade and the other queen. This occurs 36 times.
When there is one queen, conditions are met iff the other two cards are non-queen spades:
4*12*11 = 528.

So conditions are met 564 ways. The Qs is present 36+132 = 168 times, so the probability is about 31.8% that the Qs is on the board, and a further 1/3 chance that it is the middle card, so about 10.6%.