A standard deck of cards has 52 cards. There are 13 cards labeled (2,3,...,10, J
ID: 3297796 • Letter: A
Question
A standard deck of cards has 52 cards. There are 13 cards labeled (2,3,...,10, J, Q, K, A) in each of 4 suits.
i) If you deal 5 cards from a standard deck, how many different hands are possible? Assume that the order of the cards doesn’t matter for poker.
ii) A royal flush occurs when a hand contains the cards (10, J, Q, K, A) and all cards are the same suit. If you deal 5 cards at random from a standard deck, what is the probability that the hand is a royal flush?
iii) A four of a kind occurs when a hand contains the same card value in each of the four suits. If you deal 5 cards at random from a standard deck of cards, what is the probability that the hand is a four of a kind?
iv) A full house occurs when a hand contains a three of a kind and a two of a kind at the same time. For instance, (K, K, K, 4, 4) is a Full House. If you deal 5 cards at random from a standard deck of cards, what is the probability that the hand is a full house?
v) What is the probability of a 3 of a kind?
Explanation / Answer
Solution:-
i) Total number of combinations of different hands = 2,598,960
Total number of combinations of different hands = 52C5
Total number of combinations of different hands = 2,598,960
ii) The probability that the hand is a royal flush is 0.00000154.
Total combination of royal flush = 4
Total number of combinations of different hands = 52C5 = 2,598,960
The probability that the hand is a royal flush is = 4/2598960 = 0.00000154.
iii) Probability of Four of a kind = 0.000240.
This hand has the pattern AAAAB where A and B are from distinct kinds.
The number of such hands= 13C1 × 4C4 × 12C1 × 4C1 = 624
Total number of combinations of different hands = 52C5 = 2,598,960
Probability of four of a kind = 624/2,598,960 = 0.000240.
iv) Probability of getting a full house = 0.001441
This hand has the pattern AAABB where A and B are from distinct kinds.
The number of such hands = 13C1 × 4C3 × 12C1 × 4C2. = 3774
Total number of combinations of different hands = 52C5 = 2,598,960
Probability of getting a full house = 3774/2,598,960 = 0.001441.
v) Probability of three of a kind = 0.021128.
This hand has the pattern AAABC where A, B, and C are from distinct kinds.
The number of such hands = 13C1 × 4C3 ×12C2 × (4C1)2 = 54,912
Total number of combinations of different hands = 52C5 = 2,598,960
Probability of three of a kind = 54,912/2,598,960 = 0.021128.
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