# Expected value

March 10, 2010 12:46 AM

by
Elliot Frome

The phrase Expected Value has become routine in columns about video poker. What really is an Expected Value, or EV for short? How is it calculated and what does it really mean?

The expected value is the average value of all the wins attainable (after the discards are replaced), assuming that the optimum cards are retained and each unique possible draw occurs.

HUH?

Expected Value is probably best explained using an example. Let’s say you are dealt the following cards:

4♦ 4♥ 2♣ 7♥ 10♠

In this hand, there is obviously only one play, holding the pair of 4’s. What is the expected value of this hand?

Holding 2 cards, there are 16,215 ways to draw to this hand. Of these, 45 will result in 4-of-a-Kind, 165 will result in a Full House, 1,854 in 3-of-a-Kind and 2,592 will result in Two Pair. The Expected Value is calculated by multiplying the payback of a particular hand against the number of occurrences of that particular hand, and adding each possible final hand type together and dividing by the total number of hands. Assuming we are playing a full pay Jacks or Better machine, the EV is calculated as follows:

No. of |
Type |
Payback |

hands |
of hand |
of hand |

(45 |
4-of-a-kind x 25) |
plus |

(165 |
full house x 9) |
plus |

(1854 |
3-of-a-kind x 3) |
plus |

(2592 |
two pair x 2) |
divided by |

16,215 |
(total no. of hands) |
= .8236818 |

What does this .82 Expected Value really mean? First of all, any EV below 1.00 is a losing hand in the long run. Now, of course that doesn’t mean over the short run that you might not win when dealt a low pair. More than 70% of the time, a low pair will result in no payback. The remaining 28.7% will result in some form of winning hand ranging from Two Pair to Four of a Kind.

Now let’s look at a different example of a hand with 3 high cards of different suits (HON3):

J♦ Q♠ K♥ 3♣ 6♦

There are 1081 ways to draw to this 3-card hand: 32 will result in Straights, 9 in 3-of-a-Kinds, 27 in Two Pairs and 348 in a High Pairs. The EV is calculated as:

(32 x 4) + (9 x 3) + (27 x 2) + (348 x 1) / 1081 = .51

The Expected Value of this hand is considerably less than that of a Low Pair. However, with this particular hand, you will win more than 38% of the time, compared to about 29% of the time with the Low Pair.

So, what do you do if you wind up with BOTH in one hand? Suppose you are dealt:

4♣ 4♦ J♦ Q♠ K♥

What do you do? Hold the low pair or the three high cards? You’ll win more often with the 3 high cards, so do you keep those? The goal when gambling is usually to win the most coins, not the most hands. Thus, the proper play for this hand (and all others) is to play the hand with the HIGHEST Expected Value, NOT the highest win ratio. This will maximize your chance to walk away with more money than when you started. Once in a while, this strategy will have you throw away a sure winner, looking for a bigger winner: Let’s look at the following hand:

7♦ 8♦ 9♦ J♦ J♠

Do we keep the sure winner High Pair, or the 4 card inside Straight Flush (SFL4I)? The 4 card inside Straight Flush will result in a winner only about 30% of the time. The high pair is a GUARANTEED winner (once you put your coins in, even Jacks or Better/even money is considered winning). Again, we need to check the EXPECTED VALUE of each hand to determine what to do. The EV of the High Pair is 1.54. The EV of this particular 4 card inside Straight Flush (with 1 high card AND discarding a matching high card) comes to 2.38. This isn’t even close. We throw back the small fish, hoping to catch the big one, even though we will wind up with nothing more than 70% of the time.

The idea behind Expert Strategy is that each hand is played so that the cards you hold have the highest Expected Value. This may not result in the most number of hands won, but will result in the highest payback in the long run. Now, it’s not necessary to memorize the Expected Value of each hand. You simply need to remember which hand has an EV higher than the next. Most are fairly obvious, and most hands require very little decision making. Expert Strategy proves most valuable when hands ‘overlap’. We’ll cover common ‘overlapping’ hands in our next column.