Cards and the probability of drawing a specific hand

So here's a problem I'm working on...

Suppose you draw 12 cards out of a full deck of 52. I know the probability of getting any particular card is 12/52, right?

But suppose you wanted to determine the probability of a specific hand? Say, of the 12 cards drawn, you want three each of hearts, clubs, spades, and diamonds. Would I then have to add the following probabilities together: 3/52 + 3/52 + 3/52 + 3/52? It doesn't look right to me, but I'm not sure what I'm missing, exactly. Any ideas?

Re: Cards and probability

lulu said:

But suppose you wanted to determine the probability of a specific hand? Say, of the 12 cards drawn, you want three each of hearts, clubs, spades, and diamonds.

Click to expand...

The number of combinations of N choosing k is \(\displaystyle \left( {\begin{array}{c}
N \\
k \\
\end{array}} \right) = \frac{{N!}}{{k!\left( {N - k} \right)!}}\)

The answer to your question is \(\displaystyle \frac{{\left( {\begin{array}{c}
{13} \\
3 \\
\end{array}} \right)^4 }}{{\left( {\begin{array}{c}
{52} \\
{12} \\
\end{array}} \right)}}\)

Thank you, that makes much more sense! I was struggling with it, and I knew somehow I had to account for the 13 cards in each suit, and the number of the drawn cards, but my brain couldn't wrap around how to do it. Thank you again!

Are you opening a casino, lulu?

Me? Opening up a casino? Not with my luck :wink: