You are dealt a 5 card poker hand.

Question

What is the probability that all are of the same suit (all hearts, all clubs, all spades, or all diamonds)?

Answer 1

You can answer this question by the following steps:

1. Understanding the all possible outputs:

2. Calculate the probability for each output and then find the total probability.

 

P(all hearts) = P(1st card is heart) * P(2nd card is heart) * P(3rd card is heart) * P(4th card is heart) * P(5th card is heart)

= (13/52)*(12/51)*(11/50)*(10/49)*(9/48)

 

The processes to find P(all clubs), P(all spades), P(all diamonds) are similar to find P(all hearts)

But, notice that the number of hearts, clubs, spades, and diamonds are equal to each other, 13 cards.

Therefore, you just multiply the P(all hearts) by 4, you will get the P(5 cards are same suit).

 

Hope it helps.

Quan.

Answer 2

The answer is the number of ways of selecting five cards from any single suit, namely ₁₃C₅, the number of favorable cases, divided by the number of ways of dealing 5 cards from a poker deck, namely ₅₂C₅, the result is

(13×12×11×10×9)/(52×51×50×49×48)  or about .000495198

for any individual suit.  That any of the 4 suits be the suit of the flush is 4 times that.  Thank you Quan P.