A flush question

[BadPenguin]

Slimeywater's post about how good a two-pair hand you need to beat a random two-pair hand made me wonder about the same question for other hands. Suppose there's a game where you get dealt 5 cards from among the deck's 13 spades and your opponent gets dealt 5 cards that are all hearts. Without looking at his cards he bets $1; you can either fold or call.

What's the minimum hand you would call with? In other words, how big a flush do you need to have to have >50% chance of beating a random flush?

Without grinding the numbers I would guess that you'd want to call with any ace-high flush and some but not all king-high flushes. Maybe anything better than K-T-x-x-x?

[elitegimp]

I think grinding out the numbers is the way to do it:

Straight flushs: 10
A-high flush: 12 choose 4 - 2 (2 straight flushes: A-5 and T-A) = 493
K-high: 11 choose 4 - 1 = 329
Q-high: 10 choose 4 - 1 = 209
J-high: 9 choose 4 - 1 = 125
T-high: 8 choose 4 - 1 = 69
9-high: 7 choose 4 - 1 = 34
8-high: 6 choose 4 - 1 = 14
7-high: 5 choose 4 - 1 = 4
6-high: 4 choose 4 - 1 = 0 (the only 6-high flush is 65432, a straight flush)

From here you can find the midpoint, but I gotta run.

[Siegmund]

[ QUOTE ]
how big a flush do you need to have to have >50% chance of beating a random flush?

Without grinding the numbers I would guess that you'd want to call with any ace-high flush and some but not all king-high flushes. Maybe anything better than K-T-x-x-x?

[/ QUOTE ]

A good guess. Out of 1277 possible hands with no pair, #639 is K-J-T-4-2.

[AaronBrown]

Since you did all the hard work, I'll finish the job. Using the same methods, the cut-off is between KJT74 and KJT73. 640 flushes beat KJT74, 1 ties and 646 lose to it. If you go to KJT73, 644 flushes beat it, 1 ties and 643 lose to it.

[Siegmund]

Editing time has expired, so I have to put this in a new post:

I see, after reading AaronBrown's reply, that the original poster didn't explicitly exclude straight flushes - so, if those are in, we do need hand #644 and not #639. (But my list says that's KJT62, not KJT74.)

[BadPenguin]

My original post was ambiguous regarding straight flushes; assuming those are allowed as possibilities then as Siegmund says we are looking for hand #644, which I also have as KJT62.

I find it a bit surprising how high the median is - I initially guessed KT*** or better, and would think that most people's first guess would probably be lower than that, and likely include all the king-high flushes.

[AaronBrown]

You're both right. I did it by brute force, listing all 1287 flushes. It's between K J T 6 3 and K J T 6 2.

It is a surprisingly good hand. However if we change the rules and deal both flushes from the same suit (that is take all the spades and deal two five card hands), there's another surprising (to me, anyway) result. For any flush, there are 56 possible flushes your opponent can have. 325 of the 1,287 flushes are the nuts, no flush can beat them (the worst is A 9 4 3 2, that's better than a 2 3 4 5 6 straight which beats only 52 other flushes). 125 are hopeless, they can't beat anything (the best of them is J T 9 8 2, the other flush is either 7 6 5 4 3 or has a Q or better).

The median flush would beat 28 flushes and lose to 28. But there isn't one that does anything close to that. 503 flushes can beat 52 or more of the possible other hands, 455 can beat 6 or fewer. In between are 329 flushes that beat between 18 and 21 other hands. There's nothing that beats between 21 and 52, and nothing between 6 and 18.

[AaronBrown]

One more wrinkle (since I went to the trouble of generating all the hands). If you deal two hands from a standard deck, and there are no pairs in either hand, then then median hand is somewhat higher than between two flushes. Again there is a gap, K Q 4 3 2 beats 51.1% (399,768 out of 782,867) of other non-paired hands, while K J T 9 8 beats only 49.6% (388,324).