Card based probabilities

 

Counting example:

• Poker - all players are given 5 cards and must make the best possible card combos they can. The 5 cards are dealt from a standard 52 card deck.

• How many possible poker hands are there?

• Use Binomial Coefficient to find the answer

• 2,598,960 combos

• What would the probability be of getting a pair? (2 of the same value, and any 3 other cards that do not match)

• Choose 1 of the 13 values of card (13 choose 1)

• AND then choose 2 of the 4 suits (4 choose 2)

• (13 1) (4 2)

• AND Choose three of the remaining 12 values (12 choose 3)

• AND Choose a suit for each of those (4 choose 1 AND 4 choose 1 and 4 choose 1)(doing it three times allows for any suit to be chosen)

• (12 3) (4 1)3for the number of possible successes

• 1,098,240 ways or about 42%

• Could use the binomial distribution to guess the likelihood of another player having a pair

 

Cards have more information and values than dice

• Suits, number order etc.

 

When calculating the probabilities of card-like game data structures, we must also include the extra information in our counting

 

Video Game Hypothetical (Rewards System)

• Deck of cards for a reward deck

• Each card has a reward

• Duplicate values, but still unique rewards

• Each time a card is drawn from the reward deck that card is removed forever from it (Non replacement probability)

• Probability formula to help with non replacement probability

• Hyper-Geometric Distribution Formula

• The formula has 4 variables:

• The population size (N) (Number of cards in a deck for our example)

• The sample size (n) (How many cards from the rewards deck we draw)

• The number of possible successes (K)

• The number of successes we’re looking for (k)

• P = (K k) (N - K over n-k) over (N n )

Example: Magic the Gathering

• 60 cards in a deck

• 24 are island resource cards (lands)

• Start with 7 cards in hand

• Need to draw at least 4 islands by turn 4 (draw 1 card every turn)

• N is 60

• n is 11 ( start with 7, draw 4)

• K is 24

• k is [4,5,6,7,8,9,10,11]

• 8 probabilities to calculate for k

• Roughly 73% chance of drawing at least 4 cards by turn 4