Poker Hand: Probability that Five Cards are of the Same Suit

Problem
In a 5-card poker hand, what is the probability that all 5 are of the same suit?

Solution

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For 1 Particular Suit:

Approach (1) ~ 5 draws: 1 card per draw
$P_1 = \dfrac{13}{52} \times \dfrac{12}{51} \times \dfrac{11}{50} \times \dfrac{10}{49} \times \dfrac{9}{48} = \dfrac{33}{66,640}$

Approach (2) ~ 1 draw: 5 cards in 1 draw
$P_1 = \dfrac{^{13}C_5}{^{52}C_5} = \dfrac{33}{66,640}$

For 4 suits
$P = 4P_1 = \dfrac{33}{16,660}$ ← answer

Information
In poker hand, cards of the same suit and in any order is called Flush. Example of flush is (2♦, 5♦, 6♦, 9♦, Q♦ ~ a diamond flush). A flush whose cards are in sequence (i.e. 4♣, 5♣, 6♣, 7♣, 8♣) is called Straight Flush. A straight flush whose cards are composed of (10, J, Q, K, Ace) is called Royal Flush. Example of royal flush is (10♥, J♥, Q♥, K♥, A♥). Royal flush is the best possible hand in poker.

Poker Hand: Probability that Five Cards are of the Same Suit

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