A hand came up the other evening where I held three kings in five card draw, and was bet all-in by Anthony. I called to discover that he already had a pat hand. I elected to draw two, throwing away a seven and four and drawing a deuce, and then another deuce to win. Naturally he was devastated by what he considered a fluke of good luck. A couple days afterward, Anthony said he had discussed it with Aaron and they agreed that statistically I had made the wrong move by drawing two instead of holding one. I couldn’t quite follow the logic involved in why they thought I should have held on to one card or the other, but I can show that statistically speaking, I made the right decision, not knowing what pat hand he held. I think their argument was based on the idea that taking the extra draw at the king wasn’t worth the risk that I would draw a seven or four and then have to match it or still get the king, because anecdotally speaking people don’t draw the quads that often. I can break it down mathematically, and hopefully my technical writing is good enough that everyone can follow along. I should stipulate that when trying to do this in my head I got utterly confused, and I could only work it out when I sat down with paper and a calculator.
When holding KKK74, and discarding the 4, there are four outs left in the deck to improve the hand. There are three sevens and one king. This gives the hand an 8.51% to improve on the draw (4/47).
When holding the same hand and discarding both the seven or the four, there is a 1/47 chance of drawing the last king on the first draw, or 2.13% chance. That 2.13% chance is the first piece of the possible positive outcomes.
There is a 12.77% chance, or six cards in 47, that the first draw will be a seven or a four. When this occurs, there are three cards left in the deck that will improve the hand on the second draw. There are the remaining two sevens or fours and the last king. Three in forty-six times 12.77% equals .83%. That .83% is the second piece of the positive outcome.
On the first draw there are forty cards that are not a seven and not a four and not a king, or 85.1%. When one draws one of these forty cards, which I did when I drew that first deuce, there are four cards remaining in the deck that will improve the hand on the second draw, the last king, and the other three of the rank drawn, in my case the three deuces. Four in 46 times 85.1% equals 7.40%. This is the last piece of the positive outcomes when drawing two.
Notice that all the positive outcomes are mutually exclusive, and they cover all the ways it is possible to improve the hand. We can therefore add them together to get the total chance to improve. It is 2.13% plus .83% plus 7.40% equals a total of 10.36% chance, which is an improvement of 1.85% over only drawing one. Of course this isn’t much, and I might even be accused of splitting hairs, but its actually more than twenty percent better than only drawing one (1.85/8.51). When facing a pat hand with a draw left, I want the best chance I can get.
I would like to note that a lot of the debate was over the decreased percentage of getting the boat when you threw both cards away because what if they matched one you already had!?! Was it worth it to get that extra ultra slim chance at the king? Well, with 2.13 % chance of drawing that long-shot king, and we see it makes me 1.85% better to try, it would seem logical that what we give up on the boat likelihood is the difference, about .28%, and THAT is pretty negligible.
The question remains, "What made me do it right at the time, when it was difficult to figure it out when pondering on it?" I guess I just looked at that extra shot at the quads. Four of a kind didn't seem so remote to ME when I already had three of them.
Disclaimer: There are certainly very good reasons for only drawing one when the situation is changed somewhat, for instance when you suspect the card you hold is not in the other person's hand, or when you wish to create the illusion that your hand is weaker than it actually is. This particular scenario occurred during a game of three draw, and I think I had already seen a seven. It didn't occur to me to hold the four because I had not seen one. I knew when I drew the first deuce that I had NOT seen a deuce before, so I was pretty happy about my chances at another one at that point. Also, facing a player who was all-in and with a pat hand, the only illusion I was worried about was the one in my own head right before I called that said my three kings were so good.
When holding KKK74, and discarding the 4, there are four outs left in the deck to improve the hand. There are three sevens and one king. This gives the hand an 8.51% to improve on the draw (4/47).
When holding the same hand and discarding both the seven or the four, there is a 1/47 chance of drawing the last king on the first draw, or 2.13% chance. That 2.13% chance is the first piece of the possible positive outcomes.
There is a 12.77% chance, or six cards in 47, that the first draw will be a seven or a four. When this occurs, there are three cards left in the deck that will improve the hand on the second draw. There are the remaining two sevens or fours and the last king. Three in forty-six times 12.77% equals .83%. That .83% is the second piece of the positive outcome.
On the first draw there are forty cards that are not a seven and not a four and not a king, or 85.1%. When one draws one of these forty cards, which I did when I drew that first deuce, there are four cards remaining in the deck that will improve the hand on the second draw, the last king, and the other three of the rank drawn, in my case the three deuces. Four in 46 times 85.1% equals 7.40%. This is the last piece of the positive outcomes when drawing two.
Notice that all the positive outcomes are mutually exclusive, and they cover all the ways it is possible to improve the hand. We can therefore add them together to get the total chance to improve. It is 2.13% plus .83% plus 7.40% equals a total of 10.36% chance, which is an improvement of 1.85% over only drawing one. Of course this isn’t much, and I might even be accused of splitting hairs, but its actually more than twenty percent better than only drawing one (1.85/8.51). When facing a pat hand with a draw left, I want the best chance I can get.
I would like to note that a lot of the debate was over the decreased percentage of getting the boat when you threw both cards away because what if they matched one you already had!?! Was it worth it to get that extra ultra slim chance at the king? Well, with 2.13 % chance of drawing that long-shot king, and we see it makes me 1.85% better to try, it would seem logical that what we give up on the boat likelihood is the difference, about .28%, and THAT is pretty negligible.
The question remains, "What made me do it right at the time, when it was difficult to figure it out when pondering on it?" I guess I just looked at that extra shot at the quads. Four of a kind didn't seem so remote to ME when I already had three of them.
Disclaimer: There are certainly very good reasons for only drawing one when the situation is changed somewhat, for instance when you suspect the card you hold is not in the other person's hand, or when you wish to create the illusion that your hand is weaker than it actually is. This particular scenario occurred during a game of three draw, and I think I had already seen a seven. It didn't occur to me to hold the four because I had not seen one. I knew when I drew the first deuce that I had NOT seen a deuce before, so I was pretty happy about my chances at another one at that point. Also, facing a player who was all-in and with a pat hand, the only illusion I was worried about was the one in my own head right before I called that said my three kings were so good.
posted by Brinton at 2:05 PM
8 Comments:
Just some additional notes in case they come up, about what if I had seen his straight before drawing to my hand.
If his straight contains one of my two hole cards, I am 1.27% better off drawing both than holding the other, 10.79-9.52.
If his straight contains both my hole cards, I am a whopping 3.77% better drawing two for obvious reasons, 10.91%-7.14%.
If his straight contains neither of my cards, I am only slightly btter off drawing two, 9.64%-9.52%, which is .12%.
The only place it behooves me to actually hold on to the one of my cards is if his straight actually contains a king. Drawing one I hit 7.14% of the time, compared to 6.27% of the time if I draw two.
HOwever... you should not have been able to see his hand before you drew. That was a mistake on our part.
Also, he didn't have a straight. He had a flush.
I say that not because of any odds or anything. In hindsight I don't think anybody should be allowed to draw again after the cards are turned up.
Do you also think that if a player is all-in against one opponent pre-flop in Hold'em and they both show their cards there should be no cards dealt to the board? One way or the other, goingin blind, I got an extra 1.85% better off drawing two, and the range was from .12% better if he held neither a seven or a four, up to 3.77% better if he held one of each.
As for what he had, I couldn't remember, but whether a flush or a straight it doesn't make any difference. The only thing that made a difference was that no king was part of his hand.
I also find it odd that you were rooting for Anthony so much that you even say that it was "a mistake on our part." Jeez.
please visit.... dickhead.blogspot.com
thank you
No, Doyle Brunson. In Hold-em, no player has any control over their hand after they are all-in. There are five cards coming on the board, period. But in a draw game, you control whether or not you drew one or two. Of course, in all your genius, I know you know that. Please visit dickhead.blogspot.com.
And trust me, I was there and I know you put absolutely no thought into how many cards you drew. There was none of this mathematical genius posturing, so it all seems kind of irrelevant to me.
By "our" I meant all three of us.
Well, I haven't checked out your link yet, but I will AS SOON as I get this posted. As I said in the original post, the only thing I thought about was the extra draw at the king. You're absolutely right. The mathematical posturing didn't start until I heard three or four days later that I was still being second-guessed on the hand. I'm not saying that I somehow calculated all this crap out in the eight tenths of a seconds it took to decide if I was drawing one or two. I just went with my gut. I only sat down to do it for you mathematically because the logic that was explained to me about why I made a mistake seemed to very very bad, but at the same time, it opened the debate. Now, that all this "posturing" has been done, I don't think that there is a debate. Sorry I pissed you off with the "our" comment. I guess I just misunderstood you, and where the hell is Anthony in all this anyway?
A couple more quick things:
I can't seem to get anything to come up for dickhead.blogspot.com.
I'm not much of a five card draw player. I've just never played it much. There are many nuances to the game I don't know. But one thing I just never questioned was that you draw two to three of a kind unless there is a chance to deceive your opponent. Mine was all-in. I know this in the same way that I not to play 72o in Hold'em. It was just the way I was taught. Sometimes there are just rules.
If I had not seen the hand, as originally demonstrated, I would have been a fool to draw one. Having seen the hand, I would have been an even bigger fool to draw one. It didn't change anything.
I've always been of the opinion that when you only have one opponent, you're within your rights to show him as much as you want of your hand as long as you make it clear you are not folding. Maybe that isn't proper etiquette, but it should be a house rule if it isn't allowed. I think that Anthony wanted to show me that I was beaten. I don't blame him. It's a good way to get a little extra juice out of beating someone's hand, by showing them you already have the goods, and they were foolish. I don't think calling with three kings is foolish, but he wanted to show me his hand. That's undeniable. Should I be punished for that?
To cap things off, you're right, there is a difference. There are some things that I would do different having seen his hand, depending on what he had and what I held. But with three kings, would it have mattered? As long as he has me beaten, I'm drawing two. I'm not debating splitting openers to go for the flush here. I've already got three kings. The reason you didn't see me think much about whether I drew one or two was because there wasn't even a decision to make. Keep the goods. Throw out the crap. Hope for the best.
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