I have a deck with 52 cards where I miss the following cards:♤2 ♤King ♡King ♢3 ♧5 and I need to :
a) find the possibility to get a four of a kind;
b) if another card goes missing, what card should go missing to maximize the chance to get a four of a kind?
Answer depends on the type of poker your are playing. Assuming this is 5-card poker, the probability of observing a specific sequence of cards is 1/(47*46*45*44*43). A specific sequence does take order into account. Given your missing cards, you cannot make 2, K, 3, and 5 four-of-a-kind combinations, so you have the total of 9 possible ways to select a repeating card. There are also ${5 \choose 4}$ ways to select positions for your 4 cards in a sequence, and 4! ways to arrange the order in these positions. Finally, there are 43 ways to select the last card. Therefore, you get the formula: $$\frac{1}{47*46*45*44*43}*9*{5 \choose 4}*4!*43 \approxeq 0.00025 $$ Regarding b, you have to be more specific. If we are just talking about maximizing the probability of this individual combination, then you can simply choose any of the remaining 2, K, 3 or 5: the formula above can be reduced to $\frac{1}{47*46*45*44}*9*{5 \choose 4}*4! $, and removing one of these cards reduces the denominator (46*45*44*43) while keeping the rest of the formula the same. Any other card will also reduce the denominator, but also will eliminate one four-of-a-kind possibility. If you are talking about maximizing something like this $$\sum_{c \in combinations}\frac{\mathbb{P}[4 \ of \ kind]}{\mathbb{P}[c]}$$, it is a different story.