A colleague and I are having a hard time figuring out this probability question and was wondering if anyone could provide an explanation / insight into it.
The question is simple: what is the probability if you draw 2 cards at the same time (so without replacement) from a standard deck of 52, that one of those cards, or both, is a diamond?
He mapped out all the possible outcomes and came to $\frac{7}{16}$, by writing out the following:
DD DH DC DS
HD HH HC HS
SD SH SC SS
CD CH CC CS
Since 7 out of those 16 outcomes have a diamond in them, that is the probability. Trying mathematically though, the answer is not equivalent. We both believe that the formula is not right, but we can't figure out where the problem is.
This is the formula we've tried: $\displaystyle\frac{{13\choose{1}}{39\choose1}}{52\choose{2}}+\frac{13\choose2}{52\choose2}$; the left term is for 1 diamond, another card different, and the right term is for 2 diamonds. This sum comes out to $\frac{15}{34}$ or approximately .44, which is close, but not exact to the answer we'd expect above. Which is correct - or where is the mistake? An explanation of the discrepancy would be much appreciated.
Thanks a lot.