Glossary - Terms and Definitions

Expected value, referred to as EV, is a number which expresses the relative value of drawing to any combination of cards in a dealt hand based on the payoff card of the game being played. The correct way of playing any hand is to draw to the combination of dealt cards which has the highest EV. In many cases drawing five new cards will have a higher EV than drawing to any combination of the five original dealt cards.

Pat hands always have an EV of 1.00 or greater meaning that your wager is returned in a multiple represented by the EV. However, because a hand has an EV greater that 1.00 doesn't guarantee a win. It simply means that, on average, your wager will be returned multiplied by the EV. For example, the dealt hand of: 10h Ah 3c Kh Jh has an EV of 18.49 on a 9-6 Jacks or Better game (9 units paid for a Full House and 6 units paid for a Flush ); however, unless you draw the Qh ( Royal Flush ); any other Queen ( Straight ); another King, Ace, or Jack ( High Pair ); you simply wind up with "bragging rights" to a 1-card draw to a Royal Flush.

The EV is the decimal representation of a fraction whose numerator is the sum of all the possible credits that can be made by replacing any or none of the cards in the hand and whose denominator is the total number of possible hands resulting from replacing any or none of the cards in the dealt hand.

           number of possible credits from
             drawing all possible cards
   EV = ------------------------------------
        number of possible draw combinations

The following examples should help clarify this seemingly complex definition.

EV Example 1:

If the dealt hand is:

9c 8c 6c 4c 3c

you can see that the hand is a pat hand Flush which should be played as dealt. Therefore, the numerator for the EV calculation on a full-pay Jacks or Better game is 6 (the Flush payout for a single unit). Because there is only the one combination involved (the dealt hand), the denominator is 1. Therefore, the calculation is:

          6
   EV = ----- = 6.00
          1

EV Example 2:

Changing just the last card in Example 1 to present the following hand:

9c 8c 6c 4c 3h

results in a hand where you would intuitively want to hold the four clubs and draw one card to replace the 3h in an attempt make a Flush.

The numerator for the EV calculation is the the sum of all the possible credits that can be made by replacing any or none of the cards in the hand. Because there are 47 cards remaining in the deck - 9 clubs which would make a Flush and 38 others that would result in no payoff, the numerator for the EV calculation for this hand is 9 clubs times 6 units for each possible flush.

The denominator is the total number of possible hands resulting from replacing any or none of the cards in the dealt hand. In this case 47. Therefore, the calculation, rounded to two decimal digits, is:

        ( 9 * 6 )
   EV = --------- = 1.15
           47

EV Example 3:

This example demonstrates how a hand's expected value provides you with the criteria for how a hand should be played. Being dealt:

4h 4c 5s 6d 7c

results in a hand which can be played in one of two ways: a) you can hold four consecutive cards (discarding either 4) and draw one card in an attempt to make a Straight ; or, b) you can hold the pair of 4s and draw three cards.

Scenario A:

Drawing one card to 4h 5s 6d 7c results in a calculation containing numbers similar to Example 2. There are only eight cards, any 3 or any 8, which can be drawn that make a a paying hand - a Straight. Therefore, the EV for drawing one card on a full-pay Jacks or Better game with a payoff value of 4 for a Straight is:

        ( 8 * 4 )
   EV = --------- = 0.68
           47

Scenario B:

On the other hand, drawing three cards to 4h 4c would yield the following possible hands and their respective payoffs on a full-pay Jacks or Better game:

The EV calculation for the values in the above table is:

        13,356
   EV = ------ = 0.82
        16,215

Therefore, the correct way to play the hand is to draw three cards to the pair of 4s (EV = 0.82) rather than drawing one card to 4h 5s 6d 7c try to make a Straight (EV = 0.68).