Combinatorial Probability Question - Poker hand of exactly 3 suits

In summary, the probability of having exactly three suits in a five card hand dealt from a standard 52-card deck is 4 times the probability of choosing one each of spades, diamonds, and clubs, and then choosing two more cards from the remaining deck without any hearts, because the "absent" suit can be selected in 4 ways and the resulting events are mutually exclusive with the same probability.
  • #1
Sasha12
1
0

Homework Statement



A five card hand is dealt from a standard 52-card deck. What is the probability that there will be exactly three suits in the hand?

Homework Equations





The Attempt at a Solution


I started with (4 C 3)(39 C 5), but this gives me all possible hands with three or less suits. How do I get rid of the the possibility of 2 or 1 suits? I tried (4 C 3)(39 C 5) - (4 C 2)(26 C5), but I am not certain that gets rid of all possible overlaps between the sets.
 
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  • #2
so you do mean for instance club-club-heart-heart-spade?

in this case I would presume these are the options:
1. first choose 3 out of 4 suits. 3 out of 4 gives 4 possibilities.
2. for instance, say we have chosen club,heart and spade.
then there are 6 possibilities: cl-cl-cl-heart spade
cl-cl-heart-heart-spade
cl-heart-heart-heart-spade
cl-cl-heart-spade-spade
cl-heart-heart-spade-spade
cl-heart-spade-spade-spade

we have thus 4*6 possibilites= 24.
now divide this by the total amount of possibilites. I believe this is 4^5.
 
  • #3
Sasha12 said:

Homework Statement



A five card hand is dealt from a standard 52-card deck. What is the probability that there will be exactly three suits in the hand?

Homework Equations


The Attempt at a Solution


I started with (4 C 3)(39 C 5), but this gives me all possible hands with three or less suits. How do I get rid of the the possibility of 2 or 1 suits? I tried (4 C 3)(39 C 5) - (4 C 2)(26 C5), but I am not certain that gets rid of all possible overlaps between the sets.

I think it is easier to look at what is not in the hand. Say we have no hearts. There must be at least one each of spades, diamonds and clubs; you can represent this as (i) first choose exactly one each of spades, diamonds and clubs; (ii) then, from the remaining deck of 49 cards (13 hearts and 36 others) we must choose two more cards but no hearts. The probability you want is 4 times what you get here, because the 'absent' suit can be selected in 4 ways, and the 4 resulting events are mutually exclusive with the same probability.

RGV
 
Last edited:
  • #4
better listen to ray, my analysis is a bit wrong
 

1. What is combinatorial probability?

Combinatorial probability is a branch of mathematics that deals with the study and calculation of the likelihood of events occurring in a finite sample space, where the outcomes are dependent on the arrangement or combination of elements.

2. What is a poker hand?

A poker hand is a set of five cards drawn from a deck of 52 cards. In most poker games, the hand is ranked according to the standard poker hand rankings, with the highest-ranked hand winning the game.

3. What does "exactly 3 suits" mean in a poker hand?

"Exactly 3 suits" in a poker hand means that out of the five cards in the hand, three of them belong to one suit and the remaining two cards belong to two different suits. For example, a hand with 3 cards of hearts, 1 card of clubs, and 1 card of spades would be considered a poker hand of exactly 3 suits.

4. How do you calculate the probability of getting a poker hand of exactly 3 suits?

To calculate the probability of getting a poker hand of exactly 3 suits, we first need to determine the total number of possible poker hands that can be drawn from a deck of 52 cards. Then, we need to determine the number of ways to get a hand with exactly 3 suits. Finally, we divide the number of ways to get a hand with exactly 3 suits by the total number of possible hands to get the probability.

5. Why is combinatorial probability important in poker?

Combinatorial probability is important in poker because it helps players understand the likelihood of certain hands occurring and make informed decisions based on the probability of their hand winning. It also allows players to calculate the odds of their opponents having a certain hand and adjust their own strategy accordingly.

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