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Poker Hand

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.


Puzzle ID:#8068
Fun:*** (2.16)
Difficulty:*** (2.67)
Submitted By:mad-adeAhu***
Corrected By:cnmne




Which set contains proportionately more flushes than the set of all possible poker hands?

(1) Hands whose first card is an ace
(2) Hands whose first card is the ace of spades
(3) Hands with at least one ace
(4) Hands with the ace of spades

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Oct 24, 2002

Nice one, though the explanation is a bit
hard to follow. You can get a little intuition
from the inclusion-exclusion principle. If there
were only two suits in the deck (say spades and hearts),
then #(hands with As OR Ah) = #(hands with As) + #(hands with Ah) -
#(hands with As AND Ah). Both of the first two terms have the normal
proportion of flushes, and the last term being subtracted contains
no flushes, so the proportion of flushes in the whole expression must be
higher than normal. You can extend this to decks with 4 suits, but the computation
gets more involved.
Oct 24, 2002

Oct 24, 2002

Nov 07, 2002

How about this:
The odds of getting a flush are the same no matter what your first card is -- the other four have to match it.
This applies for (1) Hands whose first card is an ace, (2) Hands whose first card is the ace of spades. Since the order of the cards doesn't matter, this also applies to (4) Hands with the ace of spades. As for (3) Hands with at least one ace, it's essentially the same as (1), since you'd already have weeded out any hands that have two aces in them.
Nov 07, 2002

Quax's comment is true for #1, #2, and #4,
which are all the same. But #3 is different
from #1. To see this, try looking at the number
of two-card hands that can be drawn from a
six-card deck consisting of the A, K, and Q
of hearts and spades. There are 15 possible
hands if order is not important, of which
6 are flushes (2/5ths). If order is important
(i.e., if we distinguish between As Ks and Ks As),
then there are 30 possible hands of which 12 are
flushes. If you start with the As as your
first card, there are 5 possible hands of which
two are flushes. If you just know you have
the As in your hand, you have 10 possible
order-sensitive hands, of which 4 are flushes.
All these give you the same answer (40%) for the
probability of a flush. But if you look
at all the hands that contain an ace, there
are 9 order-independent hands, of which 4
are flushes (44%). The reason there are only
9 hands, not 10 (5+5) is that AsAh is counted
both among the 5 hands that contain As and the
5 hands that contain Ah. Now the 5 hands containing
As and the 5 hands that contain Ah each have
40% flushes, so if we just added them together
without removing the duplicate, we would get a group of 10
hands that were also 40% flushes. But in removing
the duplicate to get the actual number of distinct
hands, we are removing a hand that we know is *not*
a flush, because it contains two aces. So the proportion
of flushes in the remaining group (the non-duplicated
hands) has to go up when we remove the duplicate.
The same idea applies (on a much larger scale) if
we are looking at 5 card hands from a 52 card deck, but the
numbers are much larger and harder to verify manually.
Does this make it clearer? If not, try making a list
of all the 2 card hands you can draw from that 6 card deck,
and see if it makes more sense.

Nov 18, 2002

This is my second attempt and I didn't get to the part of sloving it because it was confusing
Jan 17, 2003

it's obviously #3

just eliminate each set, one at a time.
Jun 08, 2004

I still dont get it. Since order doest matter, all 4 hands have 1 or more aces. Doesnt this make all of them the same?
Mar 30, 2005

I agree with Jake. Each hand is known to have at least one ace. Therefore, we know that each hand has an ace, regardless of position or suit. The odds that the other four cards are of the same suit are the same. If #3 (which contains at least one ace) actually has two aces, then there is no chance of being a flush.
Apr 25, 2007

Could that be any more confusing?
There is no definite answer for the question given the information.
Are we playing Omaha, holdem, 7 card, or draw. Obviously in Omaha having two Aces would be good since they would not be suited, therefore you could make two seperate flushes.
The answer of #3 makes no sense because it says at least one A. If it said hands containing only one A then it would seem to be correct.
Though hands containing five of the same suit would seem to be the answer.
And in what game does order matter? Perhaps 7 stud, in which case having the first card be any suit doesnt matter but the second card suited would be more probable in making a flush.
Jan 13, 2009

Here are a few stats.

5,108 flushes in a deck
2,598,960 five card hands in a deck
ratio .001966

1,980 flushes with an ace in them
886,656 hands with at least one ace in them
ratio .00234

778,320 hands with exactly one ace in them
ratio .002544
Jan 13, 2009

Slight correction, there are 5,148 total flushes counting the 40 straight flushes.

ratio .0019808
Mar 17, 2009

Don't let all these facts confuse you. All the hands with exactly one ace do in fact include a larger ratio of flushes than do all the hands in a deck. In fact, all the hands with at least one ace do too.

But that doesn't mean that aces attract flushes or that flushes attract aces.
Mar 20, 2009

Great puzzle. Added instantly to my favorites list.

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