I need some of you math gurus' to settle a difference of opinion for us.
During our Tue night poker game the discussion came up as which had the
greater odds in drawing three cards. Just like three card poker.
Three of a kind or a straight flush.
I said a straight flush was harder to get.
Who's right?
Can anyone figure the odds for me.
Thanks
Brad
Amarillo
Brad Riffel wrote:
I need some of you math gurus' to settle a difference of opinion for
us. During our Tue night poker game the discussion came up as which
had the greater odds in drawing three cards. Just like three card
poker.
Three of a kind or a straight flush.
I said a straight flush was harder to get.
Who's right?
Can anyone figure the odds for me.
Your buddies buy the beer next game, Brad.
Let's look at how many ways in which you can form each hand.
For 3K, for each rank (value) there are 4 ways to form a trip (one
suit missing in each case). There are 13 ranks in all, so 52 unique
ways to form a trip.
For SF, in a given suit the possible 3 card SF's start with a 3-hi SF
and run to A-hi, a total of 11. With 4 suits, that makes 44 unique
ways to form a SF.
Clearly you'll form a 3 card SF less often than a 3-of-a-kind. To
determine odds, we need to find the total number of unique 3 card
hands = (52*51*50)/(3*2*1) = 22100.
------
A brief explanation of that math: The first product (52*51*50)
represents the total number of ways that a 3 card hand can be formed
from 52 cards (52 possibilities for the 1st card, 51 for the 2nd, 50
for the 3rd).
However, this counts multiple arrangements of the same cards
separately (e.g. 5h7s9c and 7s5h9c are each counted). Since in our
counting of 3K and SF above only "unique" arrangements were counted,
this value needs to be divided by the number of possible arrangements
of 3 cards. That's the (3*2*1) or, in short, 6.
------
So, the odds of forming a trip from 3 cards is 52 in 22100. The odds
of forming a SF is 40 in 22100.
- Harry
(I'm sure someone will let me know if a slipped a cog here somewhere,
or in light of recent events, dropped a wheel 
Harry Porter wrote:
For SF, in a given suit the possible 3 card SF's start with a 3-hi
SF and run to A-hi, a total of 11. With 4 suits, that makes 44
unique ways to form a SF.
Clearly I'm a little challenged with it comes to counting cards --
there are 12 cards in the run from 3 to A. So, there are 48 unique
ways to form a SF
------------
So, the odds of forming a trip from 3 cards is 52 in 22100. The
odds of forming a SF is 40 in 22100.
While a SF is still formed less frequently in 3 card poker than a
trip, it's closer than I suggested. The SF odds are 48 in 22100.
- H.