Can Expected Value Be Negative?

Have you heard of expected value?

It’s one of the best ways to determine how profitable or unprofitable your gambling activities are.

One question that many gamblers have is this:

Can expected value be negative?

Yes, expected value can be negative. Most gambling games have a negative expected value for players. At the same time, these games have a positive expected value for the casino or house. The house edge is another way of expressing expected value.

In the next section you can learn exactly what expected value is, how to determine the expected value of a bet or wager, and why expected value can be negative.

Contents

How Expected Value Works

Expected value is closely related to the house edge. It’s a way to analyze gambling games and individual gambling situations that show the long term average, or expectation, from the action.

If you know the house edge of a casino game, you can quickly determine the expected value for both the player and the casino, or house.

Here’s an example of expected value:

When you play baccarat and bet on the player hand, the house edge is 1.24% To determine the expected value of a bet on the player hand for the player, you multiply the 1.24% house edge times the size of the bet.

If you bet \$40 on the player hand, the expected value is – 0.496

In other words, when you make this bet you’re going to lose on 49.6 cents on average.

On the other side of this bet is the casino. The casino’s expected value is + 0.496.

Who Has the Edge?

The casino has a positive expected value, and you have a negative expected value.

The problem with understanding the expected value is when you make a bet like in the example; you either lose \$40 or win \$40. The expected value is based on a long term average.

This means that over time, your average loss on every hand of baccarat using the conditions in the example is 49.6 cents. You’re going to lose the bet slightly more often than you win, creating this average loss.

When you look at expected value from the casino’s side, they don’t make much on each bet they take, but they take a large volume of bets. This is how casinos make profits.

You can find the expected value of any bet by multiplying the house edge by the amount of the bet. Your expected value is always negative when you make a bet with a house edge.

It’s possible to make bets with an edge over the casino, and when you do this your expected value is positive. A good card counter in blackjack can play with an edge of 0.5% to 1% over the casino.

With a 1% edge over the casino, a \$50 wager has an expected value of + 0.50.

You can also make bets in poker that have a positive expected value.

And, of course, there are also bets you can make in poker that have a negative expected value.

You can learn more about expected value for casino games in the next section, and in poker in the following section.

Expected Value in Casino Games

If you know the house edge for a casino game you can determine your expected value. I’ve included a list of the house edge for the most popular casino games below.

To determine the expected value for any bet, multiply the amount of the bet times the house edge. Remember, the expected value is always negative when there’s a house edge.

Here’s a list of the house edge for popular casino games:

• Baccarat banker bet 1.06%
• Baccarat player bet 1.24%
• Blackjack 0.3% to 1% depending on the rules and strategy
• The pass line in craps 1.41%
• Craps don’t pass line 1.36%
• Craps odds bets 0%
• Roulette one zero 2.70%
• Roulette double zero 5.26%
• Slot machines 2% to 25%+
• Video poker 0.25% to 5% depending on variation, paytable, and strategy

While the expected value on a single bet might not seem like much, you usually make more than one bet. To get a better idea of how much it costs you to play your favorite casino game, determine how much you’re risking per hour and run it through the expected value calculation.

Randy provides an estimate of which casino game has the worst odds in his most recent post, too.

An Example of Expected Value per Hour

Here’s an example of expected value per hour:

You play blackjack with correct strategy and decent rules. The house edge of the game you play is .4% and you make 60 bets every hour. Your average bet size is \$60 after taking doubles and splits into account.

Multiply 60 bets times \$60 to get an hourly risk total of \$3,600. Now multiply \$3,600 times the house edge of .4% and you get an expected value of – \$14.40. In other words, you can expect to lose on average \$14.40 every hour you play.

Take a few minutes to determine the hourly expected loss for all of the games you play.

And if you’re playing one of the games with a high house edge, consider switching to one with a lower house edge to reduce your expected loss rate.

Before moving on to the section about poker, you might have noticed that there’s one bet on the list that has a 0% house edge.

The odds bet in craps has a 0% house edge, but you can’t make an odds bet until after you make a pass or don’t pass line wager.

This makes craps a good option for players that want to have a low expected loss rate. Make a table minimum bet on the pass line or don’t pass line and back it up with a full odds wager.

You can also use this calculation to help you decide how much money you should bring to the casino.

Expected Value in Poker

Poker games like Texas holdem and Omaha offer the chance to make bets that have a positive expected value. When you use the right strategies, you can play with a long term positive expectation.

Determining your expected value while you’re playing poker isn’t as easy as it is with casino games. You can look up the house edge for casino games and make a simple calculation.

But expected value in poker depends on the exact situation you’re in.

An Expected Value Example

Here’s an example of expected value in poker:

You’re playing Texas holdem and are facing a bet on the turn of \$100. The pot has \$600 in it, including the \$100 bet. You have four cards to an ace-high flush and will win the hand if you complete your flush. And you lose the hand if you miss the flush.

You can use expected value to decide if it’s better to call the bet or fold.

The deck has 9 cards in it that will complete your flush. You know the value of the 2 cards in your hand and the 4 cards on the board. This is a total of 6 cards, so the unknown cards total 46. 9 of the 46 cards complete your flush, which is a ratio of 9 to 37.

To determine the expected value of calling the \$100 bet, run the numbers for each of the unseen 46 cards. If you play this hand 46 times, the total cost is \$4,600. 46 times \$100 per hand is \$4,600.

When you hit your flush you get your \$100 back and a win of another \$600, for a total of \$700. You win nine out of the 46 hands, so the total return on a call is nine times \$700, or \$6,300.

This is higher than your investment of \$4,600, so it’s a good play to call. Take the total returns of \$6,300 and subtract the cost of \$4,600 and your total profit is \$1,700. Divide this total profit by the 46 possibilities and you get an expected value of + \$36.96.

This means that the expected value is a positive \$36.96 in this situation if you call.

Every decision you make when you play poker has an expected value. Some are negative, and some are positive. Some of them are fairly easy to determine, but some of them require some guesswork.

Winning poker players make more positive expectation plays than negative expectation plays. This is why some poker players make money in the long run and why some poker players lose in the long run.

Figure out how to use expectation when you play poker so you can make more positive expectation plays than negative expectation plays. This is the only thing you need to do to be a winning poker player.

Conclusion – Yes Expected Value Can Be Negative

Expected value is a mathematical concept that you can use to see how much a bet is expected to lose or win. In gambling situations, when one side has a positive expected value, the other side has a negative expected value.

This means that the answer is yes — expected value can be negative.

But expected value can also be positive.

1 thought on “Can Expected Value Be Negative?”

1. the negative expected value does not seem to apply to gamblers who vary their bet size, but does seem to apply to flat bettors.