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3 cards are drawn from a deck of 52 cards. Let $X$ and $Y$ represent the number of hearts and number of diamonds respectively. Find the correlation of $(X,Y)$.

I know that correlation is defined as $\frac{\mathrm{COV}(X,Y)}{\sigma_{X}\sigma_{Y}}$. I also understand that the two random variables are not independent.

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    $\begingroup$ Where are you stuck? What approaches have you tried? $\endgroup$
    – whuber
    Feb 13, 2014 at 19:24

1 Answer 1

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One (annoying) obstacle (with little pedagogical value) in problems like this is carrying out the combinatorial calculations. You are going to need information about the bivariate distribution of $(X,Y),$ which is tantamount to knowing how many possible deals of $3$ cards have $X$ hearts and $Y$ diamonds for all possible values of $X$ and $Y$. Due to the symmetry--there are as many hearts as diamonds in a deck--you only need to compute the values for $Y\ge X$, which gives six nonzero numbers. Here they are:

X   Y   Number of hands
0   0    2600
0   1    4225
0   2    2028
0   3     286
1   1    4394
1   2    1014
------  ---------------
Total:  22100

For instance, $2600$ is the number of hands with no hearts or diamonds. Therefore they are drawn from the 26 remaining cards. There are $\binom{26}{3} = 2600$ such draws. As another example, $4225$ is the number of hands with one diamond and no hearts and it equals the number of hands with one heart and no diamonds. In either case, one card comes from a 13-element set of cards and the other two come from a 26-element set of cards (all the non-heart, non-diamond cards). The total number of combinations therefore equals $\binom{13}{1}\binom{26}{2} = 4225.$ Similar techniques will yield the other four numbers.

At this point all you need to do is apply the definition of correlation to this distribution. That has some pedagogical merit in confronting you with the details (which can be illuminating) and confirming whether you have truly understood the definition.

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