Pat Hands in 5- card draw

In draw poker without the joker, what is the median pat hand, if we dont regard four of a kind as a pat hand? This should be straightforward, but i find it very difficult to calculate anyway...

[Tom Bayes]

Well, there are 52 choose 5=2,598,960 possible 5 card hands that you can be dealt. I'll cheat on the computations and refer to http://www.math.sfu.ca/~alspach/comp18/

There are 10,200 straights, 5108 flushes, 3744 full houses, 624 four-of-a-kinds (I am going to count these as pat hands, although somebody with pat quads will probably draw 1 to trick you), and 40 straight flushes, for a total of 19716 pat hands that are a straight or better, 19092 if we disregard quads.

So we see that a little over half of the pat hands are straights, so the median pat hand is going to be a fairly strong straight. There are 10 choices (A,K,Q,J,T,9,8,7,6,5) for the top card in the straight. So counting quads, the median pat hand would be approx. a king-high straight if my rough back-of-the-envelope calculations are correct.

Just as an aside, I once folded a pat hand in a pot-limit draw tournament. UTG opened with a raise, I had a pat ten-high straight and raised the pot. The next player (an extremely weak-tight passive player who virtually never raises-the kind of player who check-calls high trips) re-raised the pot. UTG calls the 2 pot-sized raises, putting himself all-in. I figure there are at least one, if not two, pat hands against me and I put the rock on at least a high flush. I fold. UTG draws two and shows down unimproved AAA. Rock shows Ace-high flush.

[MarkGritter]

[ QUOTE ]
In draw poker without the joker, what is the median pat hand, if we dont regard four of a kind as a pat hand? This should be straightforward, but i find it very difficult to calculate anyway...

[/ QUOTE ]

According to this chart,
http://www.poker1.com/mcu/tables/Table12.asp
the pat hands excluding four of a kind are:

Straight flush: 40 hands
Full house: 3744 hands
Flush: 5108 hands
Straight: 10200 hands

total = 19092, median = 9546

That puts us somewhere close to the top of the straights.
Each straight has 1020 different suits (4^5 - 4, we can't count the straight flush). So straights #1-1020 are all AKQTJ, which is where the median falls.

Bonus answer:

The 75th percentile is hand #4773, somewhere inside the flushes, flush #989 to be exact (maybe I'm off by one.)
Its identity is somewhat trickier to calculate, because the flushes have a more finely-ordered structure.

A-high flushes: 4 * (12C4 - 2) = 4 * 493 = 1980
So we're somewhere in this range.
AK-high flushes = 4 * (11C3 - 1) = 4 * 164 = 656 (#1-656)
AQ-high flushes = 4 * (10C3) = 480 (#657-1136)
So #989 is the 333rd AQ-high flush.
AQJ flushes = 4 * (9C2) = 144 (#657-800)
AQT flushes = 4 * (8C2) = 112 (#801-923)
AQ9 flushes = 4 * (7C2) = 84 (#913-996)
So #989 is the 77th AQ9-high flush.
AQ98x flushes = 4 * 6 = 24 (#913-936)
AQ97x flushes = 4 * 5 = 20 (#937-956)
AQ96x flushes = 4 * 4 = 16 (#957-972)
AQ95x flushes = 4 * 3 = 12 (#973-984)
AQ943 flushes = 4 (#985-#988)
AQ942 = flush #989-993, so this should be the 75th percentile pat hand.