I want to calculate some probabilities of the game Pinochle. But since it's difficult to translate from german into english, I'll simplify it with other words)
Basically, a deck has
- $48$ cards
- $4$ colors (red, blue, green, yellow) to each $12$ cards ($2 \times$ Ace, $2 \times$ Ten, $2 \times$ King, $2 \times$ Upper, $2 \times$ Lower, $2 \times$ Seven)
There are $3$ players. Player A, B and C. Each player receives $14$ cards and the last $6$ cards are for the tap.
What are the probabilities that
- $a)$ Player A has both red upper and blue lower in his hands
- $b)$ Player B has at least one green seven in his hand
- $c)$ Player C has exactly 5 yellow cards in his hand
- $d)$ The tab only contains Tens
This is what I have, but I don't think they are correct:
$a)$ There are 8 "upper" and 8 "lower" cards in total and there are exactly $2$ red uppers and $2$ blue lowers in the card. $\frac{2 \cdot \binom{12}{2}}{\binom{48}{12}} = 0.000000002 $
$b)$ There are exactly $2$ green sevens in the deck. $\frac{1}{48}$
$c)$ There are 12 yellow cards in total. $\frac{12}{48} \cdot \frac{11}{47} \cdot \frac{10}{46} \cdot \frac{9}{45} \cdot \frac{8}{44} = 0.00046253469$
$d)$ The tap consists of only 6 cards. There are 8 "Tens" in total. $\frac{8}{48} \cdot \frac{7}{47} \cdot \frac{6}{46} \cdot \frac{5}{45} \cdot \frac{4}{44} \cdot \frac{3}{43} = 0.0000022817$
Any help is appreciated!