Expected value is not the most likely result

Expected value is a probability-weighted average of possible results. A most likely result has the highest individual probability. A positive expected value can accompany a loss as the most likely outcome.

The difference matters when a proposal offers one attractive average in place of the range of things that could happen.

One project, three possible endings

Consider an invented one-week project. It costs $100 at the start, covering all project payments, including payment for the work. At the end of the week it receives either $0, $200 or $1,000. There are no other receipts or payments in this before-tax model.

For the exercise, assign probabilities of 65 percent to the $0 receipt, 25 percent to $200 and 10 percent to $1,000. Exactly one occurs, and these probabilities cover all the supplied possibilities. They are assumptions for the illustration, not estimated chances for a real project.

Subtracting the opening $100 gives three possible net project cash results: negative $100, positive $100 or positive $900.

Original hypothetical results and supplied probabilities for one project

Scroll across to see every column. Use the arrow keys when the table is focused.

Add the weighted amounts: negative $65 plus $25 plus $90 gives an expected value of positive $50. The OpenStax definition calculates the mean of a discrete probability distribution this way: multiply each possible value by its probability, then add.

Nothing in the supplied project pays out a net $50. That figure is the weighted average. The $900 result pulls it above zero even though the project loses $100 in the outcome assigned the highest probability.

Three answers that should stay separate

Expected net cash result

Positive $50, using every supplied result and its probability.

Most likely individual result

Negative $100, with a 65 percent probability.

Probability of a positive result

35 percent: the 25 percent and 10 percent possibilities combined.

Calling the project “likely to make $50” would collapse those answers into a claim the model does not support. It is more likely to lose money than to finish positive, and its weighted average is still positive.

The mean also does not describe how much loss someone can absorb. This project needs the opening $100 before any receipt occurs. Its expected $50 cannot fund that payment or promise to replace the money after one attempt.

Change a probability without changing the amounts

Keep all three net cash results. Reduce the probability of the $900 result from 10 percent to 5 percent, move the removed five percentage points to the negative $100 result and leave the positive $100 result at 25 percent. Calculate the new expected value before reading on.

The revised probabilities are 70, 25 and 5 percent. Their weighted amounts are negative $70, positive $25 and positive $45. They sum to zero. The most likely result remains negative $100. The probability of finishing positive is now 30 percent.

The large possible receipt remains. Its smaller assigned chance removes the positive expected value. To use either calculation for a real choice, the probabilities would need evidence beyond adding to 100 percent.

The decision-and-outcome reading asks how to assess a choice once its ending is known. The selected-stories reference asks whether the examples used to judge chances represent the attempts left out. Expected value answers a narrower question: what average follows from the distribution actually supplied?

When presented with an expected result, ask to see the outcomes and their probabilities. Then identify the most likely outcome and the chance of a loss separately. For the book’s decisions and commitments, read the opening of Infinite Upside: Building Wealth With Room to Live.