Expected value is worth knowing

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While the concept of an expected value for a wager was not created by my father (Lenny Frome), I think he coined the term and probably made it far more widely used.

The concept was used for Blackjack from the time the first Blackjack strategies were introduced. But there was no need to realize that a number existed. Instead, a player just had to memorize whether he should hit/stick/double/split.

Video Poker is a bit more complex than this. When you are dealt five cards from a deck, there are 32 ways you can play this hand. The strategy is not a single action you take, but rather which of the five cards you should keep/discard.

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While many of the 32 ways can quickly be dismissed, there are often two or more than need to be compared to each other.  This comparison is done by comparing the expected value of each way and playing the one that has the highest expected value.

In order to convince players that this is how the proper strategy is developed, my father decided it was important to explain to players how an expected value is calculated for video poker. It is not necessary to memorize the actual values, but you do need to memorize the order from high to low of each hand or partial hand in order to play the right strategy.

The actual values play a part in our what to expect category, so that a player might better understand how much he might be giving up if he chooses to go against the math.

I like to start with a simple example of how an expected value is calculated. Let’s say the player is dealt the following:

7 7 7 5 2

There is no doubt how this hand should be played. But what is the expected value of a Three of a Kind and what exactly does it mean? There are 47 remaining cards in the deck.  We will be dealt two of them. This is called 47 choose 2. The order of the cards do not matter so, it simplifies down to 47 x 46 / 2 or 1,081 possible results.

Using a simple computer program, we look at the final hand from each of these 1,081 possible draws. There are three possibilities: We can draw to Four of a Kind, a Full House or stay at Three of a Kind.

Of the 1,081 draws, 46 will result in Quads, 66 in Full Houses and the rest will stay as Trips. We now sum up the payout of all these draws (46 x 25) + (66 x 9) + (969 x 3) = 4,651. We divided this value by the number of possible draws to arrive at an expected value of 4.30.

What does this number actually mean? It means that over the long run, we can expect to win an average of 4.30 units when we are dealt a Three of a Kind. In the case of Trips, this is the expected value for every Three of a Kind (except where you were dealt a Full House and choose for reasons unknown to discard the Pair).

In some cases, as we’ll see in the coming weeks, the expected value is actually an average for a category of hands, where the hands might have slightly different expected values — let’s say a 4-Card Flush.

The 4.30 is just an average. As you can see, it is impossible to actually win 4.3 units on any one hand. You will win 3, 9 or 25.

I used Trips because it is a simple example of how to calculate an expected value and because it is the obvious only way you would play this hand. But, in the coming weeks, I will also show how the expected value of Trips can be very relevant when playing a game like Double Double Bonus Poker. 

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About the Author

Elliot Frome

Elliot Frome’s roots run deep into gaming theory and analysis. His father, Lenny, was a pioneer in developing video poker strategy in the 1980s and is credited with raising its popularity to dizzying heights. Elliot is a second generation gaming author and analyst with nearly 20 years of programming experience.

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