I'm making a tabletop roleplaying game which uses a handmade deck of cards as a chance generator. All we need to know is that there are 16 cards and four suits, but:
- Each card is of two suits;
- Six cards in the deck are of two different suits;
- And one card in the deck is of the same suit twice.
Thus, our cards look like this (with Hearts highlighted as an example):
Card | Suits |
---|---|
1 | Hearts and Clubs |
2 | Hearts and Spades |
3 | Hearts and Diamonds |
4 | Hearts and Hearts |
5 | Clubs and Hearts |
6 | Clubs and Spades |
7 | Clubs and Diamonds |
8 | Clubs and Clubs |
9 | Spades and Hearts |
10 | Spades and Spades |
11 | Spades and Diamonds |
12 | Spades and Clubs |
13 | Diamonds and Spades |
14 | Diamonds and Diamonds |
15 | Diamonds and Hearts |
16 | Diamonds and Clubs |
If I choose Hearts as my suit, and turn over a Hearts card, that's called a success. If I turn over the card with both suits as Hearts, that's two successes.
I need to calculate the probability of getting at least n successes when I draw x number of cards without replacement. For instance, it is quite common to need two successes and to draw three cards. The following sets of cards would all qualify:
- One Hearts card, one Hearts card, and one other card (2 successes)
- One double Hearts card and two other cards (2 successes)
- One double Hearts card, one Hearts card, and one other card (3 successes)
- Three Hearts cards (3 or 4 successes)
My knowledge of probability maths is basic at best, and I've set up what feels like quite a challenging system, so I'm getting nowhere calculating it. I'd be grateful for the specific probability of getting 2 successes on 3 cards, but a general formula for me to calculate successes would be even better!