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 10.
ii) One of 9 different suits.

Hence, there are 90 cards in the deck with 10 ranks for each of the 9 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 a straight flush (cards go in consecutive order like 4, 5, 6, 7, 8 and all have the same suit. Also, we are assuming there is no wrapping, so you cannot have the ranks be 8, 9, 10, 1, 2)?
b) What is the probability of being dealt a straight flush?
c) How many different ways are there to get a flush (all cards have the same suit, but they don't form a straight)? Hint: Find all flush hands and then just subtract the number of straight flushes from your calculation above.

Answer 1

a) There are 15,120 different ways to get a straight flush. b) The probability of being dealt a straight flush is 0.0000066. c) There are 6,285,840 different ways to get a flush (excluding straight flushes).

Step-by-step explanation:

a) To determine the number of ways to get a straight flush, we need to consider how many different combinations of ranks and suits can form a straight flush. Since we have 10 ranks and 9 different suits, we can choose any one of the 9 suits for each rank. However, the ranks must be in consecutive order. Therefore, we have 9 choices for the first rank, 8 choices for the second rank, 7 choices for the third rank, 6 choices for the fourth rank, and 5 choices for the fifth rank. So, the total number of ways to get a straight flush is 9 * 8 * 7 * 6 * 5 = 15,120.

b) To find the probability of being dealt a straight flush, we need to divide the number of ways to get a straight flush by the total number of possible hands. The total number of possible hands is the number of ways to choose 5 cards from a 90-card deck, which is given by the combination formula: C(90, 5). Therefore, the probability of being dealt a straight flush is: P(straight flush) = 15,120 / C(90, 5) = 0.0000066.

c) To find the number of ways to get a flush (all cards have the same suit but don't form a straight), we can subtract the number of straight flushes from the total number of flushes. The total number of flushes would be the number of ways to choose 5 cards of the same suit from the 90-card deck, which is given by the combination formula: C(90, 5). Therefore, the number of ways to get a flush (excluding straight flushes) would be: C(90, 5) - 15,120 = 6,285,840.