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
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