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Expected Value in Gambling

The concept of Expected Value (EV) can be broadly utilized in gambling, but as well as in common life. It enables to optimize an outcome, or let us say, winning chances under the conditions of risk. By means of expected value we can determine long-term house advantage (or profit) of any wager in any casino game, but we can also maximize our winning chances in Poker.

The word "Expected" can be connected with other words like "Outcome" or "Return". It is necessary to emphasize that the expected value is related to a long period or a great number of random samples. The more samples – e.g. number of spins in Roulette or rounds in Poker – the more likely the outcomes will approach the expected value.

Although the calculation of the expected value contains a bit of math and statistics, its principle is quite simple (which makes it even more powerful and beautiful) as it is clearly demonstrated on the exhibits below.

We mentioned the risk in the preface, let us define it. The risk is a possibility of deviation from the expected state (outcome, value). The risk, in contrast to uncertainty, can be measured by probability, e.g. we know what the probability to lose a stake is.

The uncertainty means that we do not know what will happen (the future is uncertain) and we do not know or we are unable to determine the probabilities of events. A little more theory and we can follow the exhibits below.

You might be interested in the use of expected value and risk in business and/or investments.

What is Expected Value

The expected value (EV) is a weighted average of all possible outcomes, whereas the weights are represented by probabilities. The expected value is a mean value, not necessarily the most probable outcome! The general formula of the expected value, EV, is the following:

EV = x1p1 + x2p2 + ... + xnpn.

where EV is the expected value, x1 ... xn are possible outcomes and p1 ... pn are respective probabilities of the outcomes to happen.

The expected value can be both positive and negative. A rationally acting person makes such decisions where the expected value is positive and refuses those decisions that bring negative expected value.

However decisions with negative expected value can be admitted in case there is nothing better and we have to choose the lesser evil – we go for a decision with the least expected loss.

This strategy applies to Poker and is the key to long-term success. In terms of casino games, whereas the result depends solely on chance, the expected value is (almost) always in player's disfavor – it secures casino's long-term profit.

House Edge Determination Based on the Expected Value

The house edge comes from a difference between real and fair payout. The real payout (declared and paid out by a casino) is lower than the fair payout. The fair payout is such a one when the expected value (of a wager) is 0, in other words when casino's long-term profit is 0.

In the following exhibits we presume to bet one dollar. There are typically only two possible outcomes of a wager – either a win or a loss. The win is then one dollar times the payout and the loss is always the one dollar. We are able to determine the probabilities of winning and losing, therefore we can mark the following:

x1 = the first possible outcome = loss of 1 dollar,
p1 = probability of loss,
x2 = the second possible outcome = win (payout),
p2 = probability of win.

Exhibit 1: Expected Value of a Single Number Bet in Roulette

To remind: There are 18 red numbers, 18 black numbers and a zero in French Roulette, that is 37 numbers in total. There is an extra number in American Roulette – so called double zero – which makes the total of 38 numbers. First let us have a look and the variables and the calculation for the French Roulette:

x1 = -1 (the loss of 1 dollar),
p1 = 36/37 (probability of loss: 36 out of 37 numbers lose),
x2 = 35 (payout 35:1),
p2 = 1/37 (probability of win: only 1 number out of 37 numbers wins).

EV = x1p1 + x2p2 = (-1) × (36/37) + 35 × (1/37) = -0.0270 = -2.7%.

A player's disadvantage (i.e. negative expected value) = house advantage. In case of this single number bet, a casino has an edge over the player 2.7% – this is also the casino's long-term profit of the turnover on single number bets.

Let us see the expected value in the American Roulette with a double zero, i.e. there is "only" one extra number, but the payout remains the same 35:1 as in the French Roulette. Now 37 out of 38 numbers lose one dollar and only 1 number of of 38 numbers wins 35 dollars, so the expected value is then:

EV = (-1) × (37/38) + 35 × (1/38) = -0.0526 = -5.26%.

The house advantage of the single number bet in the American Roulette is almost twice higher than in case of the French Roulette(!) If you feel like playing Roulette, which one of them will you choose?

Exhibit 2: Expected Value of Even-money Bets in Roulette

Even-money bets are characterized by the payout 1:1. In Roulette those are red/black, even/odd and high/low numbers. The following calculation applies to the French Roulette. If you bet e.g. on the red color, then 18 numbers win and 19 numbers lose (black numbers and a zero lose).

EV = (-1) × (19/37) + 1 × (18/37) = -0.0270 = -2.7%.

Because some (especially European's) casinos allow to take back half of the bet (or they return it automatically) when a zero comes out, the expected value = player's expected loss = house advantage is only half too, thus 1.35%. Analogously we can calculate the house edge in the American Roulette, which makes 5.26% again (the bet loses instantly when zero or double zero come out).

Expected Value in Poker

The expected value in Poker works on the same principle as in the above-described exhibits. However there is one substantial difference – in Poker you can influence by your decisions whether you will win or lose in a long-term period.

The good players accept those decisions that represent positive expected value. In all situation you can calculate or estimate whether it is worth betting: it depends on how and for how much you can improve your initial hand (or card combination if you like). This way of play is referred to as money to pot ratio and it is described on the 5-Card Stud Poker page. It is an application of the expected value.

The expected value is positive if you can improve your hand (that can win the pot) for relatively low amount of money (you have to call other player's bets) – such decisions will pay off in the long run!

→ Be sure to have a look at the Poker Variance and Dollar EV Adjusted. And then you may follow up with a simple example of variance calculation (based on two coin flipping games) and its detailed explanation. And finally do not miss the Variance Calculation for 9 Player SNG.

On the other side the expected value is negative if a chance to improve one's hand (and to win the pot) is low or relatively lower than the amount of money that you have to put to the pot. You risk a lot to win a relatively small pot.

That is a wrong decision made by bad players, who risk in a disproportionate way relying on pure chance. It can lead to success short-term and you can beat a professional incidentally. However in a long period the outcomes are close to the expected value and only those can be successful who bear in mind the rule of expected value.

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