Cards and the probability of drawing a specific hand

[lulu]

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?

 

[pka]

Re: Cards and probability

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.

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 \(\displaystyle \frac{{\left( {\begin{array}{c}
{13} \\
3 \\
\end{array}} \right)^4 }}{{\left( {\begin{array}{c}
{52} \\
{12} \\
\end{array}} \right)}}\)

 

[lulu]

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!

 

[Denis]

Are you opening a casino, lulu?

 

[lulu]

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