"5 card" guts hand rankings

[Copernicus]

I thought there was a post about this recently, but couldnt find it. What are the correct hand rankings for a variation of 3 card guts where you get to choose the best 3 out of 5 (no draw).

With normal 3 card guts (you get dealt 3, period) there are 52 sets of trips and only 48 straight flushes, so there I think SFs prevail? But when you add the element of two extra cards to choose from it seems that relationship might reverse?

[crockpot]

no, it should be the same. think about it this way: split your hand into the good 3 cards and the bad 2. there are still 48 ways the good 3 can be the straight flush and 52 ways it can be trips, and the number of combinations for the bad 2 is the same for each, so there will still be more trips.

still, i believe most 3-card guts players consider three of a kind the best hand.

[Buzz]

"no, it should be the same. think about it this way: split your hand into the good 3 cards and the bad 2.”

Crock - That gets you to a pretty good approximation in this case, to within about two per cent, or so, of the true ratio, I think, but not exactly the same.

When, (counting down from aces), you get to counting the number of possible three-card-straight-flushes made by 9h8h7h in five card hands, you can see that JhTh9h8h7h and Th9h8h7hX must have already been counted.

Meanwhile, 9s9h9d9c2c contains six three-card-trip-nine hands - but I think maybe you only want to count this hand as one three-card-hand with trip nines.

If you take these duplications into account, I think you still end up with more possible trip hands than possible straight flush hands, but not quite in the ratio of 52 to 48. (I’d guess more like 51.5 to 48.5 - something like that - just a guess).

Buzz

[bigpooch]

For trips: this is just the sum of trips or quads in draw
(so it helps to remember these numbers) or of the C(52,5)
hands there are 54912+624 = 55536 of them.

For a three card sequence of the same suit (including
A23 suited):

there are 40 straight flushes (5 cards in a row)
for four cards in a row: 2x4x47+9x4x46 = 2032
for three in a row: 2x4xC(48,2) + 10x4xC(47,2) = 52264

So altogether there are 54336 of these which is clearly
less than the number of trips+ hands. The ratio of these
sequences to trips is now 0.97839 which is quite a bit
bigger than 48/52 = 0.92308

[Buzz]

Bigpooch - Elegant solution. Thank you.

54336 as the sum of straight flushes, big bob tails and little bob tails is fine. Nifty way to think about it.

One slight problem for me.

I get 59280 (rather than 55536) as the sum of quads and trips, as follows
13*48+13*4*48*47/2 =
624+58656 =
59280,
which, if correct, gets us to about 52.2/47.8 instead of 52/48 as the ratio of five card hands containing trips to five card hands containing three-card straight flushes (also known as “little bob tails.”

O.K., I see what you must have done.
The difference, 59280-55536 = 3744.
You must have left out full houses.

That slight omission does not diminish the elegance of your thinking.

Just my opinion.

Buzz

[bigpooch]

Right you are! Sorry, but I often put up posts when I am
half asleep sans coffee. Yes, I seem to forget that
full houses include trips. In any event, since 3 card
straight flushes are more rare, they are higher ranking,
aren't they? Also, I seem to remember in my youth that
guts was often played with 2 cards in the past and the
deck would not be reshuffled until it was exhausted. If
the three card version is played the same way, it seems
helpful to remember how many aces are gone and maybe to
a much lesser extent kings.

Thanks for noticing the error!

[Phat Mack]

54336 as the sum of straight flushes, big bob tails and little bob tails is fine. Nifty way to think about it.

I agree with Buzz. Very elegant.