Poker 4 Of A Kind Vs Full House
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This week’s mailbag included an excellent question about Double Double Bonus video poker.
Poker 4 Of A Kind Vs Full House Fan
My reader wanted to know: If you are dealt a Full House consisting of three 2’s, 3’s or 4’s, how much are you giving up to go for the big payday (Quads with the A234 Kicker) vs. just keeping the Full House?
- A Full House will lose to a Straight Flush and to Four of a Kind in any confrontation. The hand can also lose to a higher-ranking Full House, determined by the rank of the three matching cards.
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I know when I’m playing and this happens there is definitely a part of me that wants to go for the big payday. Depending on the order of the dealt cards, I’m almost disappointed a Full House formed instead of just a Three of a Kind.
The expected value of the Full House is pretty obvious. On full-pay Double Double, a Full House pays 10, so that is the expected value of keeping one.
To know the impact of keeping just the Three of a Kind, we need to go a bit further. We could just look at the expected value of Three of a Kind 2’s, 3’s or 4’s on our strategy table, but this takes an average all of these cases and does not take into account the specific case of starting with a Full House (i.e. discarding a Pair along the way).
So, we need to calculate the expected value of a hand that starts as a Full House, but is played at Three of a Kind. To do this we need to look at all the outcomes of the 1,081 possible draws.
There are two possible scenarios. The Pair I discard, might be 5’s-K’s, or it might be A’s-4’s (not including the rank we have three of). If we choose to discard the Pair in this latter case, we will have fewer opportunities to pick one up as a kicker and earn the extra reward for Quads.
Let’s look at the case where the discarded Pair is not a desirable kicker (i.e. 5’s-K’s). In this case we will get the results shown in the chart below.
Hand OutcomesPaysContribution
4 of a Kind – w/Kicker 12 160 1920
4 of a Kind – other 34 80 2720
Full House 67 10 670
3 of a Kind 968 3 2904
Poker 4 Of A Kind Vs Full House Fanfiction
If we sum up the values in the Contribution column, we get 8534. We then divide this by the total number of hands (1,081) to get an expected value of 7.60, which is way below that of the Full House. If we go back to the other possibility where the Pair we discard was A’s-4’s and reduces our chances to get the big Four of a Kind, we find the expected value is even lower at only 7.45. This is more than 25% below the expected value of holding the Full House.
So, that leads to the next obvious question. What about a Full House containing three Aces? Well, in this case, we find a similar spread but above the Full House at about 12.8 and 12.3, respectively, depending on whether the pair that is discarded is 2’s-4’s or not.
Poker 4 Of A Kind Vs Full House Full
So, we have our relatively simple rule. If you have a Full House with three Aces, you keep only the three Aces. All other Full Houses are left intact.
The reader who sent me this was fairly certain the proper play was to keep the Full House, but wanted to know the impact of going for the Quads. This addresses a key part of Expert Strategy – knowing what to expect.
I’m certainly not going to advocate that you play for all of these Four of a Kinds, but if there is some reason you choose to, you should know the consequences of your decision. Don’t expect your chances of winning to go up if you play this way, but I do understand if you’re coming out to play for a few days you might be more interested in one big winner than a bunch of small success stories.
Poker 4 Of A Kind Vs Full House Plans
It’s just a question of whether you are willing to throw back some small fish in order to land the whopper of a fish.
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Elliot Frome is a second generation gaming analyst and author. His math credits include Ultimate Texas Hold’em, Mississippi Stud, House Money and many other games. His website is www.gambatria.com. Contact Elliot at [email protected].
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What's Better 4 Of A Kind Or Full House
| Example drawing to | Outs | Make on turn | Make on river | Make on turn or river | |||
|---|---|---|---|---|---|---|---|
| Prob. | Odds | Prob. | Odds | Prob. | Odds | ||
| Inside straight flush; Four of a kind | 1 | 0.0213 | 46.0-1 | 0.0217 | 45.0-1 | 0.0426 | 22.5-1 |
| Open-ended straight flush; Three of a kind | 2 | 0.0426 | 22.5-1 | 0.0435 | 22.0-1 | 0.0842 | 10.9-1 |
| High pair | 3 | 0.0638 | 14.7-1 | 0.0652 | 14.3-1 | 0.1249 | 7.01-1 |
| Inside straight; Full house | 4 | 0.0851 | 10.8-1 | 0.0870 | 10.5-1 | 0.1647 | 5.07-1 |
| Three of a kind or two pair | 5 | 0.1064 | 8.40-1 | 0.1087 | 8.20-1 | 0.2035 | 3.91-1 |
| Either pair | 6 | 0.1277 | 6.83-1 | 0.1304 | 6.67-1 | 0.2414 | 3.14-1 |
| Full house or four of a kind; (see note) Inside straight or high pair | 7 | 0.1489 | 5.71-1 | 0.1522 | 5.57-1 | 0.2784 | 2.59-1 |
| Open-ended straight | 8 | 0.1702 | 4.88-1 | 0.1739 | 4.75-1 | 0.3145 | 2.18-1 |
| Flush | 9 | 0.1915 | 4.22-1 | 0.1957 | 4.11-1 | 0.3497 | 1.86-1 |
| Inside straight or pair | 10 | 0.2128 | 3.70-1 | 0.2174 | 3.60-1 | 0.3839 | 1.60-1 |
| Open-ended straight or high pair | 11 | 0.2340 | 3.27-1 | 0.2391 | 3.18-1 | 0.4172 | 1.40-1 |
| Inside straight or flush; Flush or high pair | 12 | 0.2553 | 2.92-1 | 0.2609 | 2.83-1 | 0.4496 | 1.22-1 |
| 13 | 0.2766 | 2.62-1 | 0.2826 | 2.54-1 | 0.4810 | 1.08-1 | |
| Open-ended straight or pair | 14 | 0.2979 | 2.36-1 | 0.3043 | 2.29-1 | 0.5116 | 0.955-1 |
| Open-ended straight or flush; Flush or pair; Inside straight, flush or high pair | 15 | 0.3191 | 2.13-1 | 0.3261 | 2.07-1 | 0.5412 | 0.848-1 |
| 16 | 0.3404 | 1.94-1 | 0.3478 | 1.88-1 | 0.5698 | 0.755-1 | |
| 17 | 0.3617 | 1.76-1 | 0.3696 | 1.71-1 | 0.5976 | 0.673-1 | |
| Inside straight or flush or pair; Open-ended straight, flush or high pair | 18 | 0.3830 | 1.61-1 | 0.3913 | 1.56-1 | 0.6244 | 0.601-1 |
| 19 | 0.4043 | 1.47-1 | 0.4130 | 1.42-1 | 0.6503 | 0.538-1 | |
| 20 | 0.4255 | 1.35-1 | 0.4348 | 1.30-1 | 0.6753 | 0.481-1 | |
| Open-ended straight, flush or pair | 21 | 0.4468 | 1.24-1 | 0.4565 | 1.19-1 | 0.6994 | 0.430-1 |