Question 569652
3 cards are drawn from 12 face cards of ordinary deck of 52 playing cards. Let x be the number of kings and y be the number of jacks.  Find the joint probability distribution of x and y.
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<font color = "red">

J&#9829;  Q&#9829;  K&#9829; 
J&#9830;  Q&#9830;  K&#9830;</font>
J&#9824;  Q&#9824;  K&#9824;
J&#9827;  Q&#9827;  K&#9827; 

The trinomial probability formula for non-negative integers x,y,z
where x+y+z = 3

   {{{expr(3!/(x!y!z!))(1/3)^x*(1/3)^y*(1/3)^z}}} 

or {{{expr(6/(x!y!z!3^(x+y+z)))}}}


Let z = the number of queens (i.e., non-jacks, non-kings)

Substitution in that formula gives the following trinomial
distribution:

x  y  z    p(x,y,z)
--------------------------------
0  0  3  1/27 = 0.03703704
0  1  2  3/27 = 0.11111111 = 1/9
0  2  1  3/27 = 0.11111111 = 1/9
0  3  0  1/27 = 0.03703703
1  0  2  3/27 = 0.11111111 = 1/9
1  1  1  6/27 = 0.22222222 = 2/9
1  2  0  3/27 = 0.11111111 = 1/9
2  0  1  3/27 = 0.11111111 = 1/9
2  1  0  3/27 = 0.11111111 = 1/9
3  0  0  1/27 = 0.03703703

Edwin</pre>