# Fish Story

/This week's mailbag included an excellent question about Double Double Bonus video poker. My reader wanted to know that 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 A\2\3\4 Kicker) vs. just keeping the Full House. I know that 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 that 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 a Full House. 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 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. Also, there are two possible scenarios. The Pair that I discard, might be 5's - K's, or it might be A's - 4's (not including the rank that we have three of). If we choose to discard the Pair in this latter case, we will have less 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 following results:

Hand Outcomes Pays Contribution

Four of a Kind - w/Kicker 12 160 1920

Four of a Kind - other 34 80 2720

Full House 67 10 670

Three of a Kind 968 3 2904

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 that we discard was A's - 4's and reduces our chances to get the big Four of a Kind, we find that 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. 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 that 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 that 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 that 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. It's just a question of whether you are willing to throw back some small fishes in order to land the whopper of a fish.