# Expected value (EV)

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In probability theory, the expected value (EV) of a random variable is the weighted average of all possible values a random variable can take on.

The expected value may be intuitively understood by the law of large numbers: It can be interpreted as the long-run average of the results of many independent repetitions of an experiment (e.g. a dice roll).

Example: If you roll a dice, the possible outcomes are 1, 2, 3, 4, 5 or 6 – all with equal probability of 1/6. The expected value of a dice roll is 3.5.

This example shows that the “expected value” is not a result that may be “expected” in the ordinary sense. Rolling a 3.5 with a dice (or having 2.5 children) is impossible.

In trading, we can speak of EV as the estimated value of an investment with an unknown return.

### A simple example on EV in trading

We can buy an apple for \$10. We expect the following (based on analysis or on our experience):

• A likelihood of 50% that we can sell the apple for \$16.
• A likelihood of 25% that we can sell the apple for \$12.
• A likelihood of 25% that we will not be able to sell the apple and it goes bad.

The expected revenue of the “apple deal” would be:

(50% x \$16) + (25% x \$12) + (25% x \$0) = \$8 + \$3 + \$0 = \$11.

Thus, the expected value of the transaction would be \$11 minus the \$10 that we spend for the apple. The resulting EV of \$1 is positive, indicating that doing the deal would be profitable, or “+EV”.

The return on investment in this scenario would be +10%.
In many situations, the expected value of an investment can be estimated with thorough analysis.

## Positive result versus positive expected value

Differentiating between a positive result and a positive expected result is a key skill needed for every trader.