People often have to choose between options with uncertain outcomes. One way to think about the value of different options is in terms of expected value. The expected value of an action is the sum of the value of each potential outcome multiplied by the probability of that outcome occurring. Expected value is useful for selecting between projects.

Imagine you're 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:

Suppose another drug 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 project is:

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.