How this instrument works
This calculator uses a decision-theory framing of 'risk': expected risk equals the probability that something goes wrong multiplied by the size of the loss if it does. It's the same expected-value logic insurers, investors, and engineers use to weigh choices where both the chance of a bad outcome and its severity matter — neither number alone tells the full story.
That framing matters because intuition often focuses on just one side of the equation. A 50% chance of losing $100 sounds alarming next to a 5% chance of losing $300, but multiply each through and the second option's expected risk (0.05 × 300 = 15) is actually smaller than the first's (0.5 × 100 = 50). Expected risk gives a single number that accounts for both the likelihood and the size of the downside at once, letting two very differently-shaped risks be compared on equal footing.
This is a distinct question from epidemiology-style relative risk, which compares how much more (or less) likely an outcome is in one group versus another — a ratio of probabilities, with no dollar or loss amount involved. This calculator instead answers a decision problem: given a known probability of loss and a known amount at stake for each of two options, which one has the smaller expected cost, on average, if you made the same choice many times?
- Enter how likely option 1 is to fail, as a decimal between 0 and 1, into Option 1 — probability of loss (0-1).
- Enter how much is at stake if option 1 fails into Option 1 — amount at risk if it fails.
- Enter the same two figures for the alternative into Option 2 — probability of loss (0-1) and Option 2 — amount at risk if it fails.
- Read Option 1 — expected risk and Option 2 — expected risk — each is probability × loss for that option.
- Check Lower-risk option — it reports 1, 2, or 0 for a tie, identifying whichever option has the smaller expected risk.
Worked example — a 20% chance of losing $100 vs. a 5% chance of losing $300
Enter 0.2 into Option 1 — probability of loss (0-1) and 100 into Option 1 — amount at risk if it fails. Enter 0.05 into Option 2 — probability of loss (0-1) and 300 into Option 2 — amount at risk if it fails.
Option 1 — expected risk reads 20.0000 (0.2 × 100). Option 2 — expected risk reads 15.0000 (0.05 × 300). Even though option 2's potential loss is three times larger, its much smaller failure probability makes its expected risk the smaller of the two, so Lower-risk option reports 2 — illustrating why comparing raw loss sizes alone, without weighting by likelihood, can point you toward the wrong choice.
Questions
Isn't the option with the bigger potential loss always riskier?
Not by this measure. Expected risk weights the loss by how likely it is to happen, so a big loss that's very unlikely can carry a smaller expected risk than a smaller loss that's much more likely. In the worked example above, a 5%-chance $300 loss (expected risk 15) actually beats a 20%-chance $100 loss (expected risk 20), even though $300 sounds scarier on its own.
How is this different from relative risk in epidemiology?
Relative risk compares how much more or less likely an outcome is between two groups — it's a ratio of probabilities, like 'group A is twice as likely to experience the outcome as group B,' with no dollar amount or loss size involved. This calculator instead multiplies a probability by a stake or loss amount to get an expected cost, answering a different question: given both how likely and how costly a bad outcome is, which of two options is cheaper on average?
What does it mean if the two options tie?
A tie (Lower-risk option reports 0) means both options have exactly the same expected risk — probability times loss comes out identical for both, even if the individual probabilities and loss amounts differ. In that case, expected risk alone can't distinguish them, and you'd need another factor — such as your own tolerance for a rare-but-large loss versus a common-but-small one — to choose between them.
Does a lower expected risk guarantee a better outcome in a single decision?
No — expected risk describes the average outcome if you faced the same choice many times, not the guaranteed result of any one decision. The option with lower expected risk can still, in a single instance, produce the worse outcome purely by chance; expected value comparisons are most reliable when you're making similar decisions repeatedly, or when you're comparing to inform many independent choices rather than a single irreversible one.