Calculate an Expected Value

Expected value (EV) is a concept employed in statistics to help decide how beneficial or harmful an action might be. Knowing how to calculate expected value can be useful in numerical statistics, in gambling or other situations of probability, in stock market investing, or in many other situations that have a variety of outcomes. To calculate an expected value, you need to identify each outcome that may occur in the situation and the probability or chance of each outcome’s occurrence.

## Method 1 of 3:Learning to Find any Expected Value

##### 1. Identify all possible outcomes

Identify all possible outcomes. Calculating the expected value (EV) of a variety of possibilities is a statistical tool for determining the most likely result over time. To begin, you must be able to identify what specific outcomes are possible. You should either list these or create a table to help define the results.

For example, suppose you have a standard deck of 52 playing cards, and you want to find the expected value, over time, of a single card that you select at random. You need to list all possible outcomes, which are:

Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, in each of four different suits.

#####
2. Assign a value to each possible outcome

Assign a value to each possible outcome. Some expected value calculations will be based on money, as in stock investments. Others may be self-evident numerical values, which would be the case for many dice games. In some cases, you may need to assign a value to some or all possible outcomes. This might be the case, for example, in a laboratory experiment where you might assign a value of +1 to a positive chemical reaction, a value of -1 to a negative chemical reaction, and a value of 0 if no reaction occurred.

In the example of the playing cards, traditional values are Ace = 1, face cards all equal 10, and all other cards have a value equal to the number shown on the card. Assign those values for this example.

#####
3. Determine the probability of each possible outcome

Determine the probability of each possible outcome. Probability is the chance that each particular value or outcome may occur. In some situations, like the stock market, for example, probabilities may be affected by some external forces. You would need to be provided with some additional information before you could calculate the probabilities in these examples. In a problem of random chance, such as rolling dice or flipping coins, probability is defined as the percentage of a given outcome divided by the total number of possible outcomes.

For example, with a fair coin, the probability of flipping a “Head” is 1/2, because there is one Head, divided by a total of two possible outcomes (Heads or Tails).

In the example with the playing cards, there are 52 cards in the deck, so each individual card has a probability of 1/52. However, recognize that there are four different suits, and there are, for example, multiple ways to draw a value of 10. It may help to make a table of probabilities, as follows:

1 = 4/52

2 = 4/52

3 = 4/52

4 = 4/52

5 = 4/52

6 = 4/52

7 = 4/52

8 = 4/52

9 = 4/52

10 = 16/52

Check that the sum of all your probabilities adds up to a total of 1. Since your list of outcomes should represent all the possibilities, the sum of probabilities should equal 1.

#####
4. Multiply each value times its respective probability

Multiply each value times its respective probability. Each possible outcome represents a portion of the total expected value for the problem or experiment that you are calculating. To find the partial value due to each outcome, multiply the value of the outcome times its probability.

For the playing card example, use the table of probabilities that you just created. Multiply the value of each card times its respective probability. These calculations will look like this:

1∗452=452{\displaystyle 1*{\frac {4}{52}}={\frac {4}{52}}}

2∗452=852{\displaystyle 2*{\frac {4}{52}}={\frac {8}{52}}}

3∗452=1252{\displaystyle 3*{\frac {4}{52}}={\frac {12}{52}}}

4∗452=1652{\displaystyle 4*{\frac {4}{52}}={\frac {16}{52}}}

5∗452=2052{\displaystyle 5*{\frac {4}{52}}={\frac {20}{52}}}

6∗452=2452{\displaystyle 6*{\frac {4}{52}}={\frac {24}{52}}}

7∗452=2852{\displaystyle 7*{\frac {4}{52}}={\frac {28}{52}}}

8∗452=3252{\displaystyle 8*{\frac {4}{52}}={\frac {32}{52}}}

9∗452=3652{\displaystyle 9*{\frac {4}{52}}={\frac {36}{52}}}

10∗1652=16052{\displaystyle 10*{\frac {16}{52}}={\frac {160}{52}}}

#####
5. Find the sum of the products

Find the sum of the products. The expected value (EV) of a set of outcomes is the sum of the individual products of the value times its probability. Using whatever chart or table you have created to this point, add up the products, and the result will be the expected value for the problem.

For the example of the playing cards, the expected value is the sum of the ten separate products. This result will be:

EV=4+8+12+16+20+24+28+32+36+16052{\displaystyle {\text{EV}}={\frac {4+8+12+16+20+24+28+32+36+160}{52}}}

EV=34052{\displaystyle {\text{EV}}={\frac {340}{52}}}

EV=6.538{\displaystyle {\text{EV}}=6.538}

#####
6. Interpret the result

Interpret the result. The EV applies best when you will be performing the described test or experiment over many, many times. For example, EV applies well to gambling situations to describe expected results for thousands of gamblers per day, repeated day after day after day. However, the EV does not very accurately predict one particular outcome on one specific test.

For example, when drawing a playing card from a standard deck, on one specific draw, the likelihood of drawing a 2 is equal to the likelihood of drawing a 6 or 7 or 8 or any other numbered card.

Over many many draws, the theoretical value to expect is 6.538. Obviously, there is no “6.538” card in the deck. But if you were gambling, you would expect to draw a card higher than 6 more often than not.

## Method 2 of 3:Calculating the Expected Value of an Investment

##### 1. Define all possible outcomes

Define all possible outcomes. Calculating EV is a very useful tool in investments and stock market predictions. As with any EV problem, you must begin by defining all possible outcomes. Generally, real world situations are not as easily definable as something like rolling dice or drawing cards. For that reason, analysts will create models that approximate stock market situations and use those models for their predictions.

Suppose, for this example, that you can define 4 distinct results for your investment. These results are:

1. Earn an amount equal to your investment

2. Earn back half your investment

3. Neither gain nor lose

4. Lose your entire investment

#####
2. Assign values to each possible outcome

Assign values to each possible outcome. In some cases, you may be able to assign a specific dollar value to the possible outcomes. Other times, in the case of a model, you may need to assign a value or score that represents monetary amounts.

In the investment model, for simplicity, assume you invest $1. The assigned value of each outcome will be positive if you expect to earn money and negative if you expect to lose. In this problem, the four possible outcomes therefore have the following values, relative to the $1 investment:

1. Earn an amount equal to your investment = +1

2. Earn back half your investment = +0.5

3. Neither gain nor lose = 0

4. Lose your entire investment = -1

#####
3. Determine the probability of each outcome

Determine the probability of each outcome. In a situation like the stock market, professional analysts spend their entire careers trying to determine the likelihood that any given stock will go up or down on any given day. The probability of the outcomes usually depends on many external factors. Statisticians will work together with market analysts to assign reasonable probabilities to prediction models.

For this example, assume that the probability of each of the four outcomes is equal, at 25%.

#####
4. Multiply each outcome value by its respective probability

Multiply each outcome value by its respective probability. Use your list of all possible outcomes, and multiply each value times the probability of that value occurring.

For the model investment situation, these calculations would look like this:

1. Earn an amount equal to your investment = +1 * 25% = 0.25

2. Earn back half your investment = +0.5 * 25% = 0.125

3. Neither gain nor lose = 0 * 25% = 0

4. Lose your entire investment = -1 * 25% = -0.25

#####
5. Add together all the products

Add together all the products. Find the EV for the given situation by adding together the products of value times probability, for all possible outcomes.

The EV, for the stock investment model, is as follows:

EV=0.25+0.125+0−0.25=0.125{\displaystyle {\text{EV}}=0.25+0.125+0-0.25=0.125}

#####
6. Interpret the results

Interpret the results. You need to read the statistical calculation of the EV and make sense of it in real world terms, according to the problem.

For the investment model, a positive EV suggests that over time, you will earn money on your investments. Specifically, based on an investment of $1, you can expect to earn 12.5 cents, or 12.5% of your investment.

Earning 12.5 cents does not sound impressive. However, applying the calculation to large numbers suggests, for example, that an investment of $1,000,000 would earn $125,000.

## Method 3 of 3:Finding the Expected Value of a Dice Game

##### 1. Familiarize yourself with the problem

Familiarize yourself with the problem. Before thinking about all the possible outcomes and probabilities involved, make sure to understand the problem. For example, consider a die-rolling game that costs $10 per play. A 6-sided die is rolled once, and your cash winnings depend on the number rolled. Rolling a 6 wins you $30. Rolling a 5 wins you $20. Rolling any other number results in no payout.

#####
2. Identify all possible outcomes

Identify all possible outcomes. This is a relatively simple gambling game. Because you are rolling one die, there are only six possible outcomes on any one roll. They are 1, 2, 3, 4, 5 and 6.

#####
3. Assign a value to each outcome

Assign a value to each outcome. This gambling game has asymmetric values assigned to the various rolls, according to the rules of the game. For each possible roll of the die, assign the value to be the amount of money that you will either earn or lose. Recognize that a “no payout” means you lose your $10 bet. The values for all six possible outcomes are as follows:

1 = -$10

2 = -$10

3 = -$10

4 = -$10

5 = $20 win – $10 bet = +$10 net value

6 = $30 win – $10 bet = +$20 net value

#####
4. Determine the probability of each outcome

Determine the probability of each outcome. In this game, you are presumably rolling a fair, six-sided die. Therefore, the probability of each outcome is 1/6. You may leave this probability as the fraction of 1/6 or convert it to a decimal by dividing on a calculator. The equivalent decimal is 1/6 = 0.167.

#####
5. Multiply each value times its respective probability

Multiply each value times its respective probability. Use the table of values you calculated for all six die rolls, and multiply each value times the probability of 0.167:

1 = -$10 * 0.167 = -1.67

2 = -$10 * 0.167 = -1.67

3 = -$10 * 0.167 = -1.67

4 = -$10 * 0.167 = -1.67

5 = $20 win – $10 bet = +$10 net value * 0.167 = +1.67

6 = $30 win – $10 bet = +$20 net value * 0.167 = +3.34

#####
6. Calculate the sum of the products

Calculate the sum of the products. Add together the six probability-value calculations to find the EV for the overall game. This calculation is:

EV=−1.67−1.67−1.67−1.67+1.67+3.34=−1.67{\displaystyle {\text{EV}}=-1.67-1.67-1.67-1.67+1.67+3.34=-1.67}

#####
7. Interpret the result

Interpret the result. The EV for this gambling game is -1.67. In real world terms, this means that you can expect to lose $1.67 each time you play the game. Notice that, according to the rules of the game, it is impossible to lose $1.67. Your only options for each $10 bet are to win $30, win $20, or win nothing. However, on average, if you play this game many times, you can expect the outcome to equal an overall loss of $1.67 per play.

If you play the game once, you might win $30 (net +$20). If you play a second time, you could even win again, for a total of $60 (net +$40). However, that luck is not going to continue if you keep playing. If you play 100 times, in the end you are likely to be down approximately $167.