The expected value of an act is the sum of the value of each of its possible outcomes multiplied by their probability of occurring.

Illustration

Consider a person choosing between two medical interventions. The first intervention is a drug that succeeds in 60% of cases, and that gives an extra year of healthy life when it succeeds, and has no impact if it fails.

In the case of this drug, there are only two outcomes: success and failure. So the expected value is:

(1 year of life × 60%) + (0 years of life ×40%) = 0.6 expected years of life

Consider another drug that succeeds with a 40% probability, and gives two years of healthy life when it succeeds, but causes harm equal to half a year of healthy life lost when it fails.

Then the expected benefit of this intervention is:

(2 years of life × 40%)−(0.5 years of life × 60%) = 0.5 expected years of life

Over many cases, the first drug will likely provide more years of healthy life than the second. So if they cost the same, funding the first drug would add more healthy years of life on average.

Further reading

Conley, Sean (2016) Deworming might have huge impact, but might have close to zero impact, The GiveWell Blog, July 26.
An example of research using expected value thinking.

Karnofsky, Holden (2011) Why we can’t take expected value estimates literally (even when they’re unbiased), The GiveWell Blog, August 18 (updated 25 July 2016).
A caution about taking applied expected value estimates literally.

Todd, Benjamin (2021) Expected value: how can we make a difference when we’re uncertain what’s true?, 80,000 Hours, September.
An introduction to the concept of expected value.

Tomasik, Brian (2006) Does vegetarianism make a difference?, Essays on Reducing Suffering (updated 25 January 2014).
Another example of expected value reasoning.

Wikipedia (2010) Von Neumann–Morgenstern utility theorem, Wikipedia, April 10 (updated 13 February 2021‎).
For proofs that rational agents should select projects with the highest expected value (note that this does not imply economic risk neutrality).

Related entries

alternatives to expected value theory | altruistic wager | cluelessness | credence | decision theory | moral uncertainty | risk aversion | value of information