If I am dealt Q♢ K♢ 7♠ 3♢ 5♠ and hold the Q♢ and K♢, how do I calculate the number of possible hands that yield 1 pair of Jacks or Better?

Question

For example,

If I am dealt Q$\diamondsuit$ K$\diamondsuit$ 7$\spadesuit$ 3$\diamondsuit$ 5$\spadesuit$ and hold the Q$\diamondsuit$ and K$\diamondsuit$, how do I calculate the number of possible hands that yield 1 pair of Jacks or Better?

So far, I have:

$\frac{2{3\choose 1}{41\choose 2} +2{4\choose 2}{37\choose 1}}{47\choose 3}$ = $\frac{5364}{16215}$ = .33

This is off roughly 2% according to this poker calculator. Where am I going wrong?

Answer 1

Since you have not explained your working I'm guessing a bit, but it looks like you have calculated the probability of two kinds of hand. However there are many possibilities for getting two jacks or better from your hand:

• two jacks, or two queens, or two kings or two aces;
• three twos or threes or... or queens or kings or aces;
• flush;
• full house (in two ways);
• etc etc etc.