# 3 ways to calculate expected value

The concept of expected value (EV) is used in statistics to determine how beneficial or harmful an action could be. The expected value calculation can be used in numerical statistics, in betting or other contexts involving probability, in investments in the stock market, or in other situations where there may be different outcomes. You can calculate the expected value of a situation by first identifying each possible outcome and also the probability that each will occur.

## Steps

### Method 1 of 3: Learn to Find Any Expected Value

#### Step 1. Identify all possible outcomes

You can use the calculation of the expected value (EV) of various possibilities as a tool to determine what the most likely outcome will be over time. To start with, you need to be able to identify the specific outcomes that are possible. To do this, you can make a list of all of them or draw a table to define them.

• For example, suppose you have a standard deck of 52 cards and you want to find the expected value over time of a randomly chosen card. To do this, you must make a list of all possible outcomes:

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

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

In some contexts, such as stock market investments, the expected value calculations will be based on money, while in others, such as dice games, the calculations could be straightforward numerical values. It could also be the case that you need to assign a value to some or all of the possible outcomes, as in a lab experiment. Here you could assign a value of +1 to a positive chemical reaction, a value of -1 to a negative chemical reaction, and a value of 0 to cases where no reaction occurs.

### Following the example of the deck of cards, according to the traditional values, the aces are worth 1, the face cards are worth 10 and the other cards are worth their own number. In this case, assign each card these values

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

The term "probability" refers to the likelihood of each particular value or outcome occurring. In some contexts, such as the stock market, there may be external forces that affect the odds. In these cases, you would need additional information to be able to calculate them. In the case of problems of chance, such as tossing a coin or rolling dice, probability is defined as the percentage of a certain outcome divided by the total number of possible outcomes.

• For example, if you have a fair coin, the probability of getting the "head" when flipping it is 1/2, since there is only one head and this is divided by the total number of possible outcomes (only heads or tails).
• Following the example of the cards, because the deck has 52 cards, each one has an individual probability of 1/52. However, since there are four different suits, you can get, for example, a card with a value of 10 in different ways. It may help to create a table of probabilities, such as the following:

• 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
• The sum of all the probabilities must be 1 in total, so check that it is. All possible outcomes must appear on the list, so the sum of all of them should give 1 as the result.

#### Step 4. Multiply each value by its corresponding probability

Each possible outcome represents a part of the total expected value of the problem or experiment in question. Therefore, if you want to find the partial value due to each result, you must multiply the value of the result by its probability.

• Following the example of the deck of cards, use the probability table you created. Then multiply the value of each card by its corresponding probability. This 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}}}

#### Step 5. Find the sum of the products of these multiplications

The expected value (EV) of a set of results is equal to the sum of the individual products of the multiplications of the value by the probability. Use the table or chart you have made to add the products. The result you get should be the expected value of the problem.

• Following the example of the deck of cards, to obtain the expected value, you must add the ten products. This will give you the following:

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

• VE=34052{displaystyle {text{VE}}={frac {340}{52}}}
• VE=6, 538{displaystyle {text{VE}}=6, 538}

#### Step 6. Interpret the results

The best application for EV is in situations where the test or experiment described will be performed many times. For example, in situations involving gambling, EV can be well applied to describe expected outcomes for thousands of bettors per day and repeat the calculation day after day. However, a particular result of a specific EV test cannot be predicted very accurately.

• For example, if you are drawing a card from a standard deck, the probability of getting a 2 in a specific instance is equal to getting a 6, a 7, an 8, or any other numbered card.
• Throughout many instances of drawing a card from the deck, the theoretical expected value is 6, 538. While no card in the deck has this value, if you were to gamble, you could expect to more frequently land a card with a value greater than 6.

### Method 2 of 3: Calculate the expected value of an investment

#### Step 1. Determine all possible outcomes

You can use EV calculation as a very useful tool for investing and for predicting the stock market. However, you must first define all possible outcomes, as you would for any EV-related problem. Usually, you can't define real-world situations as easily as you can when rolling dice or drawing a card. Therefore, analysts build models that estimate stock market situations and then use them to make predictions.

• For the purposes of this example, imagine that you can determine four clearly defined outcomes for your investment, such as the following:

• 1. Earn an amount equal to your investment.
• 2. Earn half of your investment.
• 3. Neither win nor lose.
• 4. Lose your entire investment.

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

In some cases, you might assign a dollar value to possible outcomes, while in others (for example, using a model), you may need to assign a value or score to the outcomes that represents a monetary amount.

• Following the example of your investment, imagine that you invest \$ 1 (to make it simpler). Each result will be assigned a positive value if you expect to make money and a negative value if you expect to lose your money. These would be the values for the four possible outcomes relative to the \$ 1 investment:

• 1. Earn an amount equal to your investment = +1
• 2. Earn half of your investment = +0, 5
• 3. Neither win nor lose = 0
• 4. Lose your entire investment = -1

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

In contexts such as the stock market, professional analysts are dedicated to determining the probability that the value of a stock will rise or fall on a given day. These probabilities usually depend on several external factors. Therefore, statisticians work together with market analysts to be able to assign reasonable probabilities to prediction models.

### Following this example, imagine that each outcome has the same probability of occurring (25%)

#### Step 4. Multiply each value by its corresponding probability

Using your list of all possible outcomes, multiply each value by its corresponding probability.

• Following this example, this is what these calculations would look like:

• 1. Earn an amount equal to your investment = +1 * 25% = 0.25
• 2. Earn half of your investment = +0.5 * 25% = 0.125
• 3. Neither win nor lose = 0 * 25% = 0
• 4. Lose your entire investment = -1 * 25% = -0.25

#### Step 5. Add up all the products

To find the VE for a particular situation, add the products of the multiplications of each possible value by its corresponding probability.

• This would be the EV for the example of investments in the stock market:

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

#### Step 6. Interpret the results

You must observe the statistical calculation of VE to be able to interpret it in real world terms according to the problem.

• Following the example of investments, obtaining a positive EV would suggest that, over time, you would earn money from your investments. Specifically, based on a \$ 1 investment, you could earn 12.5 cents or 12.5% of your investment.
• While a profit of 12.5 cents does not seem very impressive, if you apply this same calculation to the largest investments, it would suggest that by investing \$ 1,000,000, you would earn \$ 125,000.

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

#### Step 1. Study the problem well

You must have a good understanding of the problem before you can think about the possible outcomes and their respective probabilities. For example, in the case of a dice game whose cost per play is \$ 10, a six-sided dice is rolled only once and you win money depending on the number rolled. If you roll a 6 on the die, you win \$ 30; if you roll a 5, you win \$ 20; and, if you get any other number, you don't win anything.

#### Step 2. Identify all possible outcomes

This is a relatively simple gambling game, as since a single die is rolled, any roll has only six possible outcomes: 1, 2, 3, 4, 5, and 6.

#### Step 3. Assign a value to each result

In this game, an asymmetric value is assigned to each roll of the dice according to the rules of the game. Assign each possible outcome of the die a value equal to the amount of money you will win or lose. Understand that on the results where you will not make a profit, you will lose the \$ 10 bet you have made. These are the values for the six possible outcomes:

• 1 = -\$10
• 2 = -\$10
• 3 = -\$10
• 4 = -\$10
• 5 = \$ 20 in winnings - \$ 10 of the bet = + \$ 10 (net value)
• 6 = \$ 30 in winnings - \$ 10 of the bet = + \$ 20 (net value)

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

This example assumes that the die to be rolled would be a fair six-sided die. Therefore, each outcome has a probability of 1/6. You can leave it as a fraction or do the division on a calculator to convert it to a decimal number. This fraction is equal to the decimal number 0, 167.

#### Step 5. Multiply each value by its corresponding probability

Using the table of values that you have prepared for each of the possible outcomes, multiply each of them by 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 in winnings - \$ 10 of the bet = + \$ 10 (net value) * 0, 167 = +1, 67
• 6 = \$ 30 in winnings - \$ 10 of the bet = + \$ 20 (net value) * 0, 167 = +3, 34

#### Step 6. Find the sum of all the products

Add the products of the six multiplications of probability by each expected value to get the expected value for the entire game. This calculation would be as follows:

• 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}

#### Step 7. Interpret the result

This game has an expected value of -1.67, which means that you can assume that you will lose \$ 1.67 each time you play. Keep in mind, however, that the rules of the game do not allow you to lose \$ 1,67. For every \$ 10 bet you make, you can only win \$ 30, \$ 20, or nothing. However, on average, if you play many times, the result will be equivalent to an overall loss of \$ 1,67 for each time you play.