I am trying to calculate some probabilities for a card game. Players have to draw 3 cards each time and the cards must add up to a certain value for them to win - the value changes depending on the stage of the game.
Is there a formula I can use to calculate the probabilities of drawing ≥ certain values - ≥12, ≥18 and so on? Players are drawing without replacement from a standard 52-card deck, no jokers. Aces, kings, queens, and jacks do not have a numerical value, so players are effectively drawing cards with values 2-10.
There is a generating function approach, but does not seem to have a closed form.
\begin{align*} G(x,y) &= \left((1+x)^4\prod_{i=2}^{10} (1+x\, y^i)\right)^4 \end{align*}
If the sum needs to be $\ge 12$, we need to find sum of the coefficients $[x^3 y^j]G(x,y)$ where $30 \ge j \ge 12$ and divide by $\binom{52}{3}$ to get the probability.
$$\mathbb{P}(sum \ge 12) = \frac{12236}{\binom{52}{3}} \approx 0.55366515837$$
$$\mathbb{P}(sum \ge 18) = \frac{4580}{\binom{52}{3}} \approx 0.207239819005$$