a hand is a Hush if at least 5 of its cards are of the same suit. Suppose in a given game, each player is dealt a total of 7 cards. Compute the probability that a given player will obtain a flush (Note that in each case, if n=52, then we have an ordinary 52-card dedf a o For the next two questions, we will generalise the ordinary 52-card deck to an n-card deck(where we will then let n-0o). We will do this in three different as follows (I)The deck consists of n cards(where n is a multiple of 4), of which n/4 each are Clubs, Diamonds, Hearts, and Spades. For example, perhaps several ordinary decks have been mixed together (II)The deck consists of n cards(where n is a multiple of 13), of which 13 belong to each of n /13 different suits (III) The deck consists of n cards(where n is at least 44), of which 13 each are Dia- monds, Hearts, and Spades, and the remaining n-39 are Clubs 5. For each of decks(I),(II), and(III) as above, compute the probability that a given poker hand(consisting as usual of 5 cards chosen uniformly at random from the deck) will be a flush 6. Compute the limit as n - oo of each of the three probabilities in the previous question 7. Suppose for deck(III)as above, with n a multiple of 52, we consider hands consisting of 5n/52 cards(instead of 5 cards), and say a hand is a flush if all 5n /52 cards are the same suit. Compute the probability of such a hand being a flush.(You may assume for simplicity that n> 13x52, so that Club flushes are the only possible flushes. )Then compute the limit as n - oo of this probability If you have time, you may also consider the following. For all the remaining questions ve consider only an ordinary 52-card deck Recall that an ordinary poker hand is a straight if it consists of 5 cards whose face values are in succession. For example: Ace-2-3-4-5, or 3-4-5-6-7, or 8-9-10-Jack-Queen,or 10-Jack-Queen-King-Ace are all straights. (Note that it is not permitted to "go around the corner", so that e. g. Queen-King-Ace-2-3 is not a straight. 8. Compute the probability that an ordinary poker hand is a straight 9. Suppose a player already has three cards, and their face values are 4, 5, and 6, re- will then be dealt two more cards, chosen uniformly at rando from the remainder of an ordinary 52-card deck. Compute the probability that they will end up with a straight 10. Suppose a player already has three cards, and their face values are 4, 5, and 8, re-a hand is a flush if at least 5 of its cards are of the same suit. Suppose in a given game, each player is dealt a total of 7 cards. Compute the probability that a given player will obtain a flush. For the next two questions, we will generalise the ordinary 52-card deck to an n-card deck (where we will then let n → ∞). We will do this in three different ways, as follows. (Note that in each case, if n = 52, then we have an ordinary 52-card deck.) (I) The deck consists of n cards (where n is a multiple of 4), of which n/4 each are Clubs, Diamonds, Hearts, and Spades. [For example, perhaps several ordinary decks have been mixed together.] (II) The deck consists of n cards (where n is a multiple of 13), of which 13 belong to each of n/13 different suits. (III) The deck consists of n cards (where n is at least 44), of which 13 each are Dia￾monds, Hearts, and Spades, and the remaining n − 39 are Clubs. 5. For each of decks (I), (II), and (III) as above, compute the probability that a given poker hand (consisting as usual of 5 cards chosen uniformly at random from the deck) will be a flush. 6. Compute the limit as n → ∞ of each of the three probabilities in the previous question. 7. Suppose for deck (III) as above, with n a multiple of 52, we consider hands consisting of 5n/52 cards (instead of 5 cards), and say a hand is a flush if all 5n/52 cards are the same suit. Compute the probability of such a hand being a flush. (You may assume for simplicity that n > 13×52 5 , so that Club flushes are the only possible flushes.) Then compute the limit as n → ∞ of this probability. If you have time, you may also consider the following. For all the remaining questions, we consider only an ordinary 52-card deck. Recall that an ordinary poker hand is a straight if it consists of 5 cards whose face values are in succession. For example: Ace-2-3-4-5, or 3-4-5-6-7, or 8-9-10-Jack-Queen, or 10-Jack-Queen-King-Ace are all straights. (Note that it is not permitted to “go around the corner”, so that e.g. Queen-King-Ace-2-3 is not a straight.) 8. Compute the probability that an ordinary poker hand is a straight. 9. Suppose a player already has three cards, and their face values are 4, 5, and 6, re￾spectively. The player will then be dealt two more cards, chosen uniformly at random from the remainder of an ordinary 52-card deck. Compute the probability that they will end up with a straight. 10. Suppose a player already has three cards, and their face values are 4, 5, and 8, re- 2