Ask the Slot Expert: Giving up a sure thing playing NSU

I always thought it was strange that you almost always hold any four cards to a straight flush over 3-of-a-kind in NSU when you have one deuce. For example, when dealt 2♣ 6♥ 6♦ 9♦ 10♦, you would hold 2♣ 6♦ 9♦ 10♦ and go for the gusto and the straight flush and not hold the sure 3-of-a-kind (2♣ 6♥ 6♦). The Wizard of Odds' Hand Analyzer calculates the EV of holding the double inside straight flush as 1.936170 and the EV of holding the 3-of-a-kind as 1.88067.

I hate giving up the sure thing. Let's do the math to see why it's better to gamble on getting a straight flush.

First, the correct play. There are five ways we can get the straight flush: drawing one of the remaining three deuces left in the deck, or drawing 7♦ or 8♦.

We might get a consolation prize of a flush. There are 13 diamonds in the deck. We're holding three. The deuce, 7, and 8 will give us a straight flush. That leaves seven diamonds to give us a flush.

We can also get a straight if we draw a non-diamond 7 or 8. There are six of those cards left in the deck.

We can also end up with 3-of-a-kind if we draw another 6, 9, or T. There are two 6s left in the deck and three each of 9 and T, giving us a total of 8 ways to end with 3-of-a-kind.

The remaining 21 cards give us bupkis.

Now let's look at holding 2♣ 6♥ 6♦. This will be a little more complicated because we're drawing two cards and we'll have to use combinatorics.

The best outcome we can achieve is 5-of-a-kind. We can get 5-of-a-kind using the deuces and 6s left in the deck. There are five of them and we need to draw two. The combination of five things taken two at a time is 10, so there are 10 ways to get 5-of-a-kind.

We can also get 4-of-a-kind using just one of the deuces or 6s left. Again there are five of them. We only need one this time. Next we need to fill the remaining card in our hand with one of the 42 (52 minus 5 cards dealt minus 5 cards that give us 5-of-a-kind) cards left in the deck, for a total of 210 ways to get 4-of-a-kind.

We can also get a full house if we draw a pair for replacements. We don't want deuces or 6s. There are only three 9s and three Ts left in the deck. There are three ways to draw a pair of each, so we have six ways to get a full house so far. There are nine ranks that are still complete left in the deck. There are six ways to draw four things two at a time. That gives 54 more ways for a total of 60 ways to get a full house.

The worst outcome we can have is not improving our hand and ending with the 3-of-a-kind we held. Let's do this one the easy way and subtract the number of ways we've found to improve our hand (280) from 1081 (the combination of 47 things taken 2 at a time) to get 801 ways to end with 3-of-a-kind.

The chart above shows what we calculated. It lists the hands, how much they pay per coin, the number of ways we can make it, and the product of Ways times Pays. The EV of each hold option is Tot Pays/Ways from the Totals line.

As much as I don't like doing it because I end up with nothing just under half the time (21 times out of 47), it is better to hold a 4-card double inside straight flush with one deuce over a pair and a deuce in NSU. (Note: You hold the 3-of-a-kind if the 4-card straight flush is Ace low.)

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