Question

Suppose you are playing a game of five-card poker where you are
dealt a hand of 5 cards from a standard deck of 52 playing cards.
Each card has one of 13 values, 2,3,…,10,J,Q,K,A2,3,…,10,J,Q,K,A,
and one of 4 suits, ♡,♢,♠,♣♡,♢,♠,♣

A hand called a **royal flush** consists of cards of
the values 10,J,Q,K,A10,J,Q,K,A, all from the same suit. For
instance, a royal flush with hearts would be

10♡,J♡,Q♡,K♡,A♡10♡,J♡,Q♡,K♡,A♡.

Now, actually getting a royal flush in poker is incredibly rare (in
fact, it should be easy to see see that there are only 4 ways that
you can draw a royal flush). However, to people that play poker
occasionally, it seems like **almost** getting a royal
flush is all too common. It's this scenario that we'll be
interested in in this problem. We'll answer the question: "How many
ways are there to pick a 5-card poker hand that is one card away
from being a royal flush?"

For example, if you have 10♣,J♣,Q♣,K♣,A♡10♣,J♣,Q♣,K♣,A♡, the hand
is one card away from being a royal flush.

Similarly, if you have 10♢,J♢,4♣,K♢,A♢10♢,J♢,4♣,K♢,A♢, the hand is
also one card away from being a royal flush.

So we want to figure out how many such hands there are. Don't worry
- we're going to walk through this together! We can break this task
of creating an "almost-royal-flush" hand into a few
steps:

Figure out which 4 of the 5 royal flush card values we've gotten right

Figure out what is the suit of the 4 royal flush cards that we have

Figure out what the last card is - that card that ruined your royal flush!

Figure out how to combine Steps 1-3 to get our answer

**Part A.** First, completely ignoring suit, how many
ways are there to pick which 4 of the 5 royal flush card values
**are** in your hand?

---

**Part B.** Alternatively, how many ways there are to
pick which 1 of the 5 royal flush card values **is**
**not** in your hand?

---

**Part C.** Look at your answers to Parts A and B.
Notice anything? Do not proceed until you notice something
interesting.

---

**Part D.** Okay, so we've dealt with the number of
ways to get 4 out of the 5 royal flush card values. But this is an
almost-royal-flush we're talking about, so at least four of these
cards should be the same suit. How many ways are there to decide
which suit an almost-royal-flush will be?

---

**Part E**. Great! Now we have the almost-royal-flush
cards' values and suits. So those 4 cards are completely
determined, but we still need to decide what the other card should
be.

After we selected the first 4 cards, there are 48 cards left in the
deck. One of these 48 cards, however, will leave us with an actual
royal flush if we pick it. How many ways are there to choose our
fifth, and final, card such that we do not end up with a royal
flush?

---

**Part F.** The procedure of creating a
poker hand that is one card away from being a royal flush consists
of 3 steps, all of which must be completed in order to accomplish
the task. Use the appropriate rule (product rule or sum rule) to
calculate the total number of 5-card poker hands that are one card
away from being a royal flush, and your answers from the previous
parts.

---

**Part G. **How many 7-card poker hands are
one card away from containing a 5-card royal
flush? *Hint: the first two steps of this procedure
are the same as the 5-card problem; it is the last part - picking
the card(s) that ruined your royal flush - that needs to be
modified.*

Answer #1

a) The number of ways to pick which 4 of the 5 royal flush card
values **are** in your hand is (ignoring suit).

.

d) There are 4 suits. The above number should be multplied by 4.

e) The number of ways to choose our fifth, and final, card such that we do not end up with a royal flush

.

f) Use product rule to calculate the total number of 5-card poker hands that are one card away from being a royal flush, and your answers from the previous parts.

g) The total number of 7-card poker hands that are one card away from being a royal flush, and your answers from the previous parts. is found analogously,

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.

Determine the poker odds of drawing the following hand from a
standard card deck. (4 suits, 13 ranks in each suit.) What are the
odds of drawing a royal flush (AKQJ10 all of the same suit)?
Drawing cards is WITHOUT replacement. (NOTE: If necessary, lay out
52 cards on a table and do a dry run before computing the
probabilities!) In order to get a royal flush in spades, you must
pick the following 5 cards:
What is the probability...

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);...

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.

Probabilities with a deck of cards. There are 52 cards in a
standard deck of cards. There are 4 suits (Clubs, Hearts, Diamonds,
and Spades) and there are 13 cards in each suit. Clubs/Spades are
black, Hearts/Diamonds are red. There are 12 face cards. Face cards
are those with a Jack (J), King (K), or Queen (Q) on them. For this
question, we will consider the Ace (A) card to be a number card
(i.e., number 1). Then for each...

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...

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?

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

Suppose you and a friend are playing cards and you are each
dealt 3 cards. You have a 8 through 10 in your hand. You are about
to be dealt one more card. What is the probability that you are
dealt a Jack given that
(a) Your friend has no Jacks in his hand.
(b) Your friend has exactly one Jack in his hand.
Suppose you are playing Poker alone. You have four cards (6♡♡,
7♡♡, 8♡♡, and 9♡♡). You...

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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