Question

We are creating a new card game with a new deck. Unlike the normal deck that has 13 ranks (Ace through King) and 4 Suits (hearts, diamonds, spades, and clubs), our deck will be made up of the following. Each card will have: i) One rank from 1 to 17. ii) One of 6 different suits. Hence, there are 102 cards in the deck with 17 ranks for each of the 6 different suits, and none of the cards will be face cards! So, a card rank 11 would just have an 11 on it. Hence, there is no discussion of "royal" anything since there won't be any cards that are "royalty" like King or Queen, and no face cards! The game is played by dealing each player 5 cards from the deck. Our goal is to determine which hands would beat other hands using probability. Obviously the hands that are harder to get (i.e. are more rare) should beat hands that are easier to get. a) How many different ways are there to get any 5 card hand? The number of ways of getting any 5 card hand is b)How many different ways are there to get exactly 1 pair (i.e. 2 cards with the same rank)? The number of ways of getting exactly 1 pair is What is the probability of being dealt exactly 1 pair? Round your answer to 7 decimal places. c) How many different ways are there to get exactly 2 pair (i.e. 2 different sets of 2 cards with the same rank)? The number of ways of getting exactly 2 pair is What is the probability of being dealt exactly 2 pair? Round your answer to 7 decimal places. d) How many different ways are there to get exactly 3 of a kind (i.e. 3 cards with the same rank)? The number of ways of getting exactly 3 of a kind is What is the probability of being dealt exactly 3 of a kind? Round your answer to 7 decimal places. e) How many different ways are there to get exactly 4 of a kind (i.e. 4 cards with the same rank)? The number of ways of getting exactly 4 of a kind is What is the probability of being dealt exactly 4 of a kind? Round your answer to 7 decimal places. f) How many different ways are there to get exactly 5 of a kind (i.e. 5 cards with the same rank)? The number of ways of getting exactly 5 of a kind is What is the probability of being dealt exactly 5 of a kind? Round your answer to 7 decimal places. g) How many different ways are there to get a full house (i.e. 3 of a kind and a pair, but not all 5 cards the same rank)? The number of ways of getting a full house is What is the probability of being dealt a full house? Round vour answer to 7 decimal places.

Answer 1

In the custom card game with 102 cards across 17 ranks and 6 suits, we calculate the number of ways to get different hands and their probabilities using combinatorial functions. The probability is derived from the total number of possible hands, which is calculated as C(102, 5), and then dividing the specific hand combinations by this total.

Step-by-step explanation:

To solve these card game probability questions, we use combinatorics, a branch of mathematics dealing with the counting, arrangement, and combination of objects. Combinatorics and probability specifically are important for understanding the likelihood of different hands in card games. In your newly designed card game with a deck of 102 cards, where each player receives 5 cards, and there are 17 ranks across 6 suits, various hands have differing probabilities of occurring based on how rare they are. The calculation steps for each type of hand involve combinatorial functions such as combinations and permutations.

Calculations for different hands:

• Any 5 card hand: This can be calculated using the combination formula C(n, k) where n is the total number of cards, and k is the number of cards chosen. C(102, 5) gives us the total number of 5-card hands possible.
• Exactly 1 pair: First, pick one rank for the pair out of 17, then pick 2 of the 6 cards from that rank, and finally pick 3 more cards from the remaining 102 - 6 cards, ensuring they all have different ranks.
• Exactly 2 pairs: Similar approach as for 1 pair, but you must pick two different ranks and then two cards from each of those ranks before selecting the fifth card from the remaining cards.
• Exactly 3 of a kind: Choose one rank and then three cards of that rank, followed by two more cards with different ranks.
• Exactly 4 of a kind: Choose one rank and then four cards of that rank, followed by one more card of a different rank.
• Exactly 5 of a kind: Pick one rank and all five cards from that rank.
• Full house: Choose one rank for the 3 of a kind, then another rank for the pair.

Each of these calculations will lead to finding the number of ways each hand can occur, and the probability can be found by dividing this number by the total number of possible 5-card hands derived from C(102, 5).