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Best of Donald Catlin

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Those Frustrating Low Pairs

6 March 2004

I don't know about you but whenever I play a version of Jacks or Better Video Poker it seems like those low pair hands show up with an uncanny frequency and with the same uncanny frequency seem to seldom lead to anything fruitful. Yet, when I look at strategy tables they seem to rank right up there with four card straights and three card straight flushes. That is to say, the strategy tables tell us that a low pair is a decent hand. It sure never feels like that to me. Let's take a closer look.

There are 2,598,560 ways to deal a five card hand from a 52-card deck. If you want to see how one arrives at such a number you can go into the archives and look for my article Hey New York, Bring Back those Big Dippers or better still you can refer to my new book The Lottery Book, The Truth Behind the Numbers, available from Huntington Press (1-800-244-2224), Payone Press (1-800-944-0406), Barnes and Noble, or Amazon.

If you think of a deck of cards as consisting of 13 packets of 4 equally ranked cards each, then nine of these packets contain low cards. The number of ways to select one of these is obviously 9 and then there are 6 ways to choose a pair of cards from the chosen packet. In other words, there are 9 x 6 or 54 ways to choose a pair of low cards. This done we select any three of the remaining twelve packets (220 ways) and select one card from each (64 ways). Altogether, then, there are 54 x 220 x 64 or 760,320 ways to select a five-card hand consisting of a low pair. Dividing this into 2,598,960 we obtain approximately 3.42 so that on average we should see a low pair every 3.42 hands. Well that explains why we get so many of them. What happens when we do?

If we toss the three strays and keep the pair, then our remaining deck contains 47 cards consisting of a pair matching the two in our hand, three packets of three cards each corresponding to the three we tossed (9 cards), and nine packets of four equally ranked cards (36 cards).

There are 16,215 ways of dealing three cards from a 47-card deck. How many of these will give us a final hand of two pair? The answer is that we can either choose one of the four-card packets (9 ways), choose two cards from it (6 ways) and then choose one of the remaining 41 cards (we don't want to pick cards matching our original pair or the two just picked), or we can pick one of the three-card packets (3 ways), choose two from it (3 ways), and select any of the 42 remaining cards (the 36 cards in the four card packets and the six cards in the remaining two three-card packets). Hence there are 9 x 6 x 41 + 3 x 3 x 42 or 2,592 hands with two pairs. Dividing this into 16,215 we find that we will have two pair once in every 6.26 draws or about 29.25% of the time we draw to a low pair.

I'm not going to drag you through all of the remaining calculations but trips will occur 1854 times, a full house will occur 165 times (about 1% of the time), and four of a find will only occur 45 times (once every 360 draws). If we add all of these together, a winning hand will occur 4656 times out of 16,215 draws, or about 28.7% of the time.

Since a low pair occurs 29.25% of the time, and when it does a winning draw will occur 28.7% of the time, the frequency of being dealt a low pair that wins is 0.2925 x 0.287 or 0.0839. This means that approximately once in every 12 hands you'll be dealt a low pair that wins something. Now you see why getting a low pair doesn't give you that winning feeling. So why are they ranked so high?

The answer is that some of the hands that low pairs produce are big payers. Take 9/6 Jacks for example. Using the above figures we can easily construct the following table.

Hand

Frequency

Payout

Product

Two Pair

2592

2

5,184

Trips

1854

3

5,562

Full House

165

9

1,485

Four of a Kind

45

25

1,125

Losers

11,559

0

0

Totals

16,215

---

13,357

Expectation Drawing to a Low Pair

Dividing 13,357 by 16,215 we obtain approximately 0.8237. In other words, if we are dealt a low pair, the hand has an expected return (ER) of a bit over 82 cents on the dollar. Of course, it cost a dollar to play so this represents a negative expectation for the player, but it is still considerably better than, say, Jack/Queen unsuited (ER of 0.5059).

Does all of this make you feel any better about low pairs? Probably not, but at least now you know the facts. See you next month.

Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers