# How To Calculate an Expected Value

EV is a concept used 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, stock market investing, and many other situations that have a variety of outcomes.To calculate an expected value, you need to know the probability or chance of each outcome occurring.

### Step 1: Pick out all possible outcomes.

Calculating the expected value of a variety of possibilities is a statistical tool for determining the most likely result over time.Identifying what specific outcomes are possible is the first thing you need to do.You can either create a table to help define the results or list them.If you have a standard deck of 52 playing cards, you want to find the expected value over time of a single card that you pick at random.In each of the four suits, you need to list the possible outcomes: ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K.

### Step 2: Each possible outcome has a value assigned to it.

Money will be used for some expected value calculations.Many dice games have self-evident numerical values.You may need to assign a value to some outcomes.In a laboratory experiment, you could assign a value of +2 to a positive chemical reaction, a values of -1 and 0 if no reaction occurred.The traditional values for the playing cards are 1, face cards all equal 10, and all other cards have a value equal to the number shown on the card.For this example, assign those values.

### Step 3: The probability of each possible outcome should be determined.

There is a chance that an outcome will occur.Some external forces may affect the stock market’s probabilities.You would need to provide more information before you could calculate the probabilities in these examples.The percentage of a given outcome divided by the total number of possible outcomes is known as the probability.The probability of flipping a “Head” is 1/2, because there is one Head, divided by a total of two possible outcomes.There are 52 cards in the deck, so each card has a chance of being the winner.There are four different suits and multiple ways to draw a value of 10.Check that the sum of the probabilities is 16/52.The sum of probabilities should be 1 since your list of outcomes should represent all possibilities.

### Step 4: Divide each value’s probability by it.

A portion of the total expected value is represented by each possible outcome.The partial value is determined by the value of the outcome times its probability.The table of probabilities should be used for the playing card example.Divide the value of each card by its probability.The calculations will look like this.

### Step 5: The sum of products can be found.

The sum of individual products of the value times its probability is known as the expected value.Add up the products and you will get the expected value for the problem.The sum of the ten separate products is the expected value.The result will be EV=4+8+16+20+24+8+32+36+16052.

### Step 6: Take into account the result.

The best time to apply the EV is when you will be doing the described test many, many times.EV can be used to describe expected results for thousands of gamblers per day, repeated day after day.The EV is not very good at predicting one outcome on a test.When drawing a playing card from a standard deck, the chance that a 2 will be drawn is the same as if you drew a 6 or 7 or 8 card.The theoretical value to expect is 6.538.There is no “6.538” card in the deck.You would expect to draw a card higher than 6 if you were gambling.

### Step 7: Define all possibilities.

Calculating EV can be used to make stock market predictions.Define all possible outcomes of the EV problem.Rolling dice or drawing cards are things that are easy to define in a real world situation.Analysts will use models that approximate stock market situations for their predictions.You can define 4 different results for your investment.The results are: 1.You can earn an amount equal to your investment.You can earn back half your investment.Neither loss nor gain.You will lose your entire investment.

### Step 8: Values should be assigned to each possible outcome.

You may be able to assign a dollar value to the outcomes.In the case of a model, you may need to assign a value or score that represents monetary amounts.Assume you invest \$1 in the investment model.If you expect to earn money or lose, the assigned value will be positive.The values of the four possible outcomes are relative to the \$1 investment.You can earn an amount equal to your investment.If you earn back half your investment, it’s +0.5 3.No gain or loss is defined as 0 4.If you lose your entire investment, it’s -1.

### Step 9: The probability of each outcome should be determined.

Professional analysts spend their entire careers trying to figure out if a stock will go up or down on a given day.Many external factors affect the probability of outcomes.Statisticians and market analysts will assign reasonable probabilities to prediction models.Assume that the probability of each of the four outcomes is equal.

### Step 10: Divide the outcome value by its probability.

You can use your list of possible outcomes to calculate the probability of each value occurring.The model investment situation would look like this: 1.You can earn an amount equal to your investment.If you earn back half your investment, you’ll get 0.125 3.Neither gain nor lose is equal to 25%.If you lose your entire investment, it’s 0.25.

### Step 11: Add the products together.

Add the products of value times probability to find the EV for the situation.The EV for the stock investment model is 0.25.

### Step 12: Take the results into account.

According to the problem, you need to read the statistical calculation of the EV and make sense of it in real world terms.A positive EV suggests that you will make money on your investments over time.You can expect to make 12.5 cents for every \$1 invested.It doesn’t sound impressive to earn 12.5 cents.The calculation suggests that an investment of \$1,000,000 would earn \$125,000.

### Step 13: You should be familiar with the problem.

Understand the problem before thinking about possible outcomes.Consider a game that costs \$10 per play.Cash winnings depend on the number of rolls of a 6-sided die.You can win \$30 if you roll a 6.You can win \$20 if you roll a 5.Rolling any other number causes no payouts.

### Step 14: Pick out all possible outcomes.

This is a relatively easy game to play.There are only six possible outcomes if you roll one die.The numbers are 1, 2, 3, 4, 5 and 6.

### Step 15: Each outcome has a value assigned to it.

According to the rules of the game, asymmetric values are assigned to various rolls.For each roll of the die, assign a value to the amount of money you will make or lose.It’s a good idea to know that a “noPayout” means you lose your \$10 bet.The values for all six possible outcomes are as follows.

### Step 16: The probability of each outcome is determined.

You are rolling a six-sided die in this game.The probability of each outcome is 1/6.You can either leave it as the fraction of 1/6 or use a calculator to convert it to a decimal.1/6 is the equivalent of 0.167.

### Step 17: Divide each value’s probability by it.

You can use the table of values to calculate the probability of each die roll.

### Step 18: The sum of the products is calculated.

The EV can be found with the six probability-value calculations.This calculation is called EV, it’s a display style.

### Step 19: Take into account the result.

This gambling game has an EV of 1.67.You can expect to lose \$1.67 every time you play the game.It is impossible to lose \$1.67 according to the rules of the game.If you place a \$10 bet, you only have three options: win \$30, win \$20, or win nothing.If you play this game many times, you can expect the outcome to be the same as the overall loss.You could win \$30 if you play the game once.You could win again for a total of \$60 if you play again.If you keep playing, luck is not going to continue.You are likely to be down \$167 if you play 100 times.