# 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**

**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.

Subscribe to MATHalino.com on