Question

5. In poker, a “flush” is a five-card hand where all five cards have the same suit. A hand has “three of a kind” when any three cards have the same rank. “Four of a kind” is defined analogously. A flush beats a three of a kind, but is beaten by four of a kind. Demonstrate that this ranking of hands corresponds to how unlikely each hand type is to occur when drawing five cards at random (Hint: just count the number of different hands of each type, and then rely on the principle of exchangeability).

Answer #1

Number of ways of selecting 5 cards out of 52 cards is

Four of a kind:

There are total 13 denominations and each denomination has 4 cards. So number of ways of selecting 1 denominations and then 4 cards out of 4 is

And since we need 4 of same kind so remaining 1 card must come from
different denomination so number of ways of selecting 1
denominations out of remaining 12 denominations and then 1 card
from selected denomination is

So number of ways of selecting four of a kindis :

So probability of getting four of a kind:

-------------------

3 of a kind:

Number of ways of selecting 1 denominations out of 13 is C(13,1). Number of ways of selecting 3 cards out of 4 cards of selected denomination is C(4,3). And then select two denominations out of remaining 12 denominations is C(12,2) and then 1 card from each selected denominations is C(4,1)C(4,1). So number of ways are there to draw a 5 card poker hand that contains 3 a kind is

C(13,1)C(4,3)C(12,2)C(4,1)C(4,1) = 54912 ways

So probability of getting three of a kind:

--------------------

Flush:

There are total 4 suits and each suit has 13 cards. So number of ways of selecting 1 suit and then 5 cards out of 13 is

So probability of getting flush:

That is out of these it is most likely to get three of a kind, then flush and then four of a kind.

If you have played poker, you probably know some or all the
hands below. You can choose 5 cards from 52 in (52) ways. But how
many of them would be a Royal Flush or a Four-of-a-Kind?
Royal Flush: All five cards are of the same suit and are of the
sequence 10 J Q K A.
Four-of-a-Kind: Four cards are all of the same rank.

Five cards are selected from a 52-card deck for a poker hand. a.
How many simple events are in the sample space? b. A royal flush is
a hand that contains the A, K, Q, J, and 10, all in the same suit.
How many ways are there to get a royal flush? c. What is the
probability of being dealt a royal flush? Please, please, please
explain and not just give the answer or formula.

1. A five-card poker hand is dealt from a standard deck of
cards. Find the probability that:
a. The hand contains exactly 3 Clubs and exactly 1
Spade.
b. The hand contains at least two aces
c. The hand contains all cards from the same suit
d. The hand contains three cards from one suit and two cards
from different suit
e. The hand contains no more than one spade

2. Assume that you play poker with six-card hands. (A straight
has all six cards in a row; as in standard poker, aces are high or
low and there is no wrap-around. A flush has all six cards in the
same suit. (Use combinatorics)
(f) How many six-card hands contain a straight?

Part A Poker Hands: In this activity, we will apply some of the
various counting techniques that we have studied including the
product and sum rules, the principle of inclusion-exclusion,
permutations, and combinations. Our application will be counting
the number of ways to be dealt various hands in poker, and
analyzing the results.
First, if you are not familiar with poker the following is some
basic information. These are the possible 5-card
hands:
Royal Flush (A,K,Q,J,10 of the same suit);...

You read online that the probability of being dealt
four‑of‑a‑kind in a five‑card poker hand is 1/4165 . Explain
carefully what this means. In particular, explain why it does not
mean that if you are dealt 4165 five‑card poker hands, one will be
four‑of‑a‑kind. Select the best explanation from the choices.
It does mean that if you are dealt 4165 five‑card poker hands,
one will be four‑of‑a‑kind. It does not mean that all will be
four‑of‑a‑kind.
The probability is actually...

5 cards are randomly selected from a standard deck of 52
cards to form a poker hand. Determine the probability of being
dealt a straight flush (five cards in sequence in the same suit but
not a royal flush. Note: A royal flush is 10, Jack, Queen, King,
Ace all in the same suit. Note: Aces can be high or
low).

In Texas Hold’Em, at the end of the hand the player has 7 cards
available to use to get the best of the 5-card hands we calculated
in class.
a. How many 7-card hands are possible?
b. What is the probability of having a royal flush be a part of
your hand?
c. What is the probability of all 7 cards being the same
suit?
d. What are the odds of having a four of a kind? How does this...

1.How many ways are there to draw a ﬁve-card poker hand that
contains ﬁve cards of the same suit?
2. How many ways are there to draw a ﬁve-card poker hand that
contains at least one ace?

Q19. Consider an ordinary 52-card North American playing deck (4
suits, 13 cards in each suit).
a) How many different 5−card poker hands can be drawn from the
deck?
b) How many different 13−card bridge hands can be drawn from the
deck?
c) What is the probability of an all-spade 5−card poker
hand?
d) What is the probability of a flush (5−cards from the same
suit)?
e) What is the probability that a 5−card poker hand contains
exactly 3 Kings...

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