Risk Formula

Risk is the possibility of loss or negative outcome, quantified by combining the probability of the event with the severity of its impact: Expected Loss =.

The Formula

Expected Loss=P(loss)×amount of loss\text{Expected Loss} = P(\text{loss}) \times \text{amount of loss}

When to use: What could go wrong, how likely is it, and how bad would it be?

Quick Example

Investment risk: 20%20\% chance of losing $100 vs 5%5\% chance of losing $1000.

Notation

RR or Risk=P×I\text{Risk} = P \times I where PP is probability and II is impact

What This Formula Means

Risk is the possibility of loss or negative outcome, quantified by combining the probability of the event with the severity of its impact: Expected Loss = P(loss) times amount of loss.

What could go wrong, how likely is it, and how bad would it be?

Formal View

Expected Loss=∑iP(lossi)⋅Li\text{Expected Loss} = \sum_{i} P(\text{loss}_i) \cdot L_i where LiL_i is the magnitude of each potential loss

Worked Examples

Example 1

medium
A business faces two possible disasters: (A) equipment failure — probability 0.10, cost \$50,000; (B) data breach — probability 0.02, cost \$500,000. Calculate the expected loss from each and determine which poses greater financial risk.

Answer

Expected loss: Equipment = \$5,000; Data breach = \$10,000. Data breach is the greater financial risk.

First step

1
Expected loss A: EA=P(A)×cost=0.10×50000=$5,000E_A = P(A) \times \text{cost} = 0.10 \times 50000 = \$5,000

See the full worked solution + why-it-works coaching

SetupKey insightWhy it worksCommon pitfallConnection

Unlock answer keys One Family plan — every worked solution, all subjects

Example 2

hard
Should you buy insurance for \$200/year that covers a \$5000 loss occurring with probability 0.03? Calculate expected value for both choices and discuss why someone might still buy the insurance.

Example 3

medium
A bike costs $600. Theft insurance is $45/year. Last year 3%3\% of similar bikes were stolen. Compute the expected loss without insurance and decide whether the premium beats the expected loss.

Common Mistakes

  • Ranking by probability alone — multiply by the loss amount so severity counts.
  • Ranking by severity alone — a catastrophic but near-impossible event may carry little expected loss.
  • Adding probability and loss instead of multiplying — Expected Loss is P×amountP\times\text{amount}, a product.

Why This Formula Matters

Risk is why a small chance of a huge loss can outweigh a likely small one — it's the math behind insurance, safety, and everyday decisions. Looking only at probability (or only at severity) leads to consistently bad choices. Recognizing it by "Am I combining the probability of a loss with the size of that loss?" — rather than by familiar numbers — is what lets a student tell it apart from probability and expected value and severity / impact alone in a mixed problem set.

Frequently Asked Questions

What is the Risk formula?

Risk is the possibility of loss or negative outcome, quantified by combining the probability of the event with the severity of its impact: Expected Loss = P(loss) times amount of loss.

How do you use the Risk formula?

What could go wrong, how likely is it, and how bad would it be?

What do the symbols mean in the Risk formula?

RR or Risk=P×I\text{Risk} = P \times I where PP is probability and II is impact

Why is the Risk formula important in Math?

Risk is why a small chance of a huge loss can outweigh a likely small one — it's the math behind insurance, safety, and everyday decisions. Looking only at probability (or only at severity) leads to consistently bad choices. Recognizing it by "Am I combining the probability of a loss with the size of that loss?" — rather than by familiar numbers — is what lets a student tell it apart from probability and expected value and severity / impact alone in a mixed problem set.

What do students get wrong about Risk?

The procedure for risk is the easy part; the trap is ranking by probability alone. Asking "Am I combining the probability of a loss with the size of that loss?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Risk formula?

Before studying the Risk formula, you should understand: probability.

← Back to Risk See All Examples Practice Problems