Probability of different card hands

Question

I'm looking at doing a game for an upcoming expo and I'm having a hard time with the math for it. I'm wanting to lay out 25 cards (all suited the same so let's say all Hearts) and give out prizes based on what is drawn from the attendees. Each person would get to pick 5 cards. The cards would be laid out on a table in rows/columns of 5. (So like a big square)

My layers of winning would be: 1 pair, 3 of a kind, 4 of a kind, Straight, and Royal Flush.

and I'm looking at laying my cards out like so:

$$ \begin{matrix} \textbf{How Many} & \textbf{Card Type} \\ 4 & 2 \\ 4 & 3 \\ 3 & 4 \\ 2 & 5 \\ 2 & 6 \\ 2 & 7 \\ 2 & 8 \\ 1 & 9 \\ 1 & 10 \\ 1 & A \\ 1 & J \\ 1 & Q \\ 1 & K \end{matrix} $$ I need to figure out the probability of drawing each of the winning options. Ie., probability of drawing a pair from this layout. Probability of drawing a 3 of a kind from this layout. etc.

Answer 1

For what its worth, simulating one million hands yielded these estimates for the probabilities. It looks like the odds are pretty close to those for a full deck, except that the hands three-of-a-kind or better are slightly more likely with the shortened deck.

junk            48.50575 %
pair            42.37255 %
two-pair         4.77935 %
3ofkind          3.3505  %
straight         0.68395 %
fullhouse        0.22435 %
4ofakind         0.08135 %
royalstraight    0.0022  %

Found with this Python program.