Finding value in games of chance

Oct 17, 2005 3:00 AM

A term that I use very frequently in my columns is Expected Value. As far as I know, this is another term that my dad, Lenny, coined many years ago, at least where gambling is concerned. The concept existed long ago, but I’m not sure if anyone put a name to it, or bothered to calculate the specific value for different situations.

Probably one of the first places where the concept of Expected Value, or EV for short, came into being was in the game of blackjack. While many decisions to hit or stick are pretty obvious, the very close ones were decided based on which had the higher EV.

The same principles were used for deciding hands to double or split. Whether someone calculated the probabilities mathematically (very difficult) or using computer simulations (much easier), it still came down to the same basic principle: Which action will put more money in my pocket or cause me to lose less money?

When you’re dealt a 16 in blackjack and the dealer has a face card, why does virtually every strategy say hit? Yes, it’s true that the dealer is unlikely to bust, but that’s only the simple English rationale of the math behind the game of blackjack. Using a computer simulation to run 1 million hands of a 6-deck game, we find the results of the hit vs. stick decision of a 16 looking into a face card (see chart below).

Based on this simulation, the EV of hitting is .429, while the EV of sticking is .424. This is not a huge difference, only about a half-cent per dollar wagered. If you look closely, you’ll note that the player won’t actually win MORE hands by hitting. He will actually be relying on the pushes that occur as a result of hitting.

Yet, you won’t find a serious blackjack player who won’t hit the 16 looking into the face card (unless he’s card counting).

For years, blackjack players have relied on the concept of expected value. Many did not know the exact EVs for a particular hand, but they KNEW that the expected value of hitting was greater than that of sticking.

The same math applies to all blackjack hands. In fact, recently I was asked about a game called Blackjack Switch, in which the player has the opportunity to switch the second card dealt to him on each of his two hands. When asked about the strategy, I replied that you simply need the expected value of each hand. If switching increases the expected value, you switch. If it decreases the expected value, you don’t switch. Because of the options to Double Down and Split, you also need to take into account the wager size when determining the overall expected value (I’ll save this concept for a future article), but it still comes down to maximizing the expected value of each hand.

There are approximately 300 different hand combinations that can occur in the game of blackjack (from the first 3 cards being dealt, ignoring suits). When we take the probability of each combination and multiply it by the expected value of the best way to play the hand, and add up all these numbers, we wind up with the ”˜expected value’ of the entire game of blackjack, more commonly known as the payback.

Because we have maximized the expected value of each possible play, we have also maximized the overall payback of the game. Any changes in our strategy will reduce the payback.

The same concepts apply to virtually every casino game. Anytime a decision needs to be made by the Player, if the Player wants to maximize the payback of the game, he chooses the option that maximizes the expected value at that point in time. In the case of blackjack, there are usually only 2-4 options to choose from (hit, stick, double, split). In the case of video poker, there are actually 32 options for every hand, as there are 32 ways to discard 0-5 cards from a 5-card hand. Most of these options are quickly dismissed, generally leaving the player with anywhere from 1-3 choices. Fortunately, the Player does not have to start calculating the expected values in his head. Instead, he merely needs to use a strategy chart to guide him, and match one of these choices with one of the hands on the strategy chart. Picking the option with the highest expected value will allow the player to achieve the maximum payback possible from the game.