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Make sure your child knows how important |
Poker is a game played with a standard 52 playing card deck. There are 4
suits: spades, clubs, diamonds and hearts. Each suit has the same 13
cards. There are 9 numbered cards: 2 through 10, and 4 face cards: jack,
queen, king and ace.
Each player is dealt 5 cards from the deck. The object is to get a 5-
card combination (known as a hand) that is more valuable than everyone
else's. The harder it is to get a particular hand, the more valuable it
is. Here are the hands in order from least importance to most:
none the same
pair
two pair
three of a kind
full house
four of a kind
What we are doing in this exercise is figure out the different ways of getting these hands.
For starters, the total number of poker hands is
. The method I will use to figure out
the number of ways of getting the above hands I call the
choose/choose method. I call it that
because what I will do first is choose from the 13 cards (also known as
faces) common to each
suit, then choose the suit for each group of cards that have the same
face. The method for choosing is exactly the same as the
elevator problem.
None the same: First we choose 5 faces
from the 13. We have
choices there. Then,
for each face, we have 4 suits to choose 1 suit from. So, the number of
suit choices is 4x4x4x4x4 = 1,024. The total
number of hands where none of the cards match is
1,287 x 1,024 = 1,317,888. The probability of
getting this hand is 50.7%
Pair: First we choose the face for the
pair. There are 13 choices for that. Now we have 12 faces from which to
choose the remaining 3 which works out to be
combinations. So, the total number of face
choices for this case is
13 x 220 = 2,860
In choosing the suits, for the pair we have
choices and for the rest of the cards, once
again we have 4. So, the number of suit choices in this case is
6x4x4x4 = 384. Ergo, (there's some Latin for
you) the total number of hands with one pair is
2,860 x 384 = 1,098,240. The probability of
getting this hand is 42.3%
THAT MEANS THE TOTAL PROBABILITY OF GETTING AT BEST A PAIR IS A WHOPPING 93%!!! Makes you wonder how people won at poker in those old spaghetti western movies, doesn't it? (unless they got a hand like a straight or royal flush which we cover later)
Two pair: We have to choose 2 faces, one
for each pair. We have
choices there. Then,
we have 11 faces left to choose from for the last card. So, the number
of face choices for this case is
78 x 11 = 858
As far as the suit choices go, just like the last case, we have 6
choices for the first pair, 6 for the second pair, and 4 for the single
card. The number of suit choices is
6x6x4 = 144. The total number of hands with
two pair is 858 x 144 = 123,552. The probability
of getting this hand is 4.8%
Three of a kind: The smarter ones among
you should be able to do this one in your sleep:
choices for the 3 cards that are the
same
choices for the other 2 cards
So there are 13 x 66 = 858 card choices in
this case.
To choose the suits, we have
for the 3 of a
kind cards, and 4 choice for each of the other 2 cards. The total number
of suit choices is 4x4x4 = 64. So, the total
number of hands for this case is
858 x 64 = 54,912. The probability of
getting this hand is 2.1%
Full house: Do you think point form will
do?
choices for the 3 cards that are the
same
choices for the 2 cards that are the
same
Total number of card choices:
13 x 12 = 156
suit choices for the 3 of a kind cards
choices for the 2 of a kind cards
Total number of suit choices:
4 x 6 = 24
Total number of hands that are a full house:
156 x 24 = 3,744
Probability of getting this hand for real: 0.144%
Probability of getting this hand in a spaghetti western: happens all the
time
Four of a kind: Deja vu, anyone?
choices for the 4 cards that are the
same
choices for the 1 card left
Total number of card choices:
13 x 12 = 156
suit choices for the 4 of a kind cards
choices for the last card
Total number of suit choices:
4
Total number of hands that have 4 of a kind:
156 x 4 = 624
Probability of getting this hand for real: 0.024%
Believe it or not, this is not the most popular hand in gambling movies.
More on this later.
Anyway, if you add all these numbers up, they add up to 2,598,960.
Since I'm such a nice guy, I'll let you figure out how I came up with these numbers
Straight: 5 cards in a row. The suits
don't matter.
Number of card combinations: 10
Number of suit combinations: 1,024
Total: 10,240
Probability: 0.39%
Note: The reason there are 10 combinations is because the ace can be
either low or high.
Flush: 5 cards from the same suit. The
cards don't matter.
Number of card combinations: 1,287
Number of suit combinations: 4
Total: 5148
Probability: 0.2%
Note: Some argue that the total should subtract 40 from the
total of 5148 since they consider the straight flush a completely
different hand than the flush. Whatever.
Straight flush: 5 cards in a row from
the same suit.
Number of card combinations: 10
Number of suit combinations: 4
Total: 40
Probability: you figure it out
Royal flush: The 10, jack, queen, king
and ace from the same suit.
Number of card combinations: 1
Number of suit combinations: 4
Total: 4
Probability: in the real world or Hollywood?