What does probability measure?
Probability describes uncertainty about an event. It lies between 0 and 1. A probability of 0.20 means a 20% chance under the stated information and model. It does not promise exactly 20 occurrences in every set of 100 observations.
An experiment produces an outcome. The sample space contains all possible outcomes. An event is a subset of that space. For a loan over the next year, a simple space is {default, no default}. Define the horizon before assigning probabilities.
Four rules at a glance
| Rule | Expression | Meaning |
|---|---|---|
| Complement | P(not A) = 1 − P(A) | If default probability is 2%, survival probability is 98%. |
| Addition | P(A or B) = P(A) + P(B) − P(A and B) | Subtract the overlap to avoid counting it twice. |
| Conditional probability | P(A given B) = P(A and B) / P(B) | Update the reference group to outcomes where B occurs. Requires P(B) > 0. |
| Multiplication | P(A and B) = P(A given B) × P(B) | This rule works even when the events depend on one another. |
Independence versus mutually exclusive events
Independent events do not change one another’s probabilities: P(A and B) = P(A)P(B). Mutually exclusive events cannot both occur, so their intersection has probability zero. Two mutually exclusive events with positive probabilities are therefore not independent.
Two borrowers in the same industry can default together during a downturn. Assuming independence simply because the borrowers are different can understate joint default risk.
Worked example: a warning signal and default
Consider 10,000 hypothetical loans. The annual default rate is 2%. A warning signal identifies 80% of defaults, but also flags 10% of loans that do not default. All figures are teaching assumptions.
| Outcome | Flagged | Not flagged | Total |
|---|---|---|---|
| Default | 160 | 40 | 200 |
| No default | 980 | 8820 | 9800 |
| Total | 1140 | 8860 | 10000 |
The 80% detection rate is P(flag given default). It differs from the 14.04% probability of default after seeing a flag. The initial default rate matters because non-defaulting loans greatly outnumber defaulting loans.
Bayes’ theorem and total probability
For the loan example, the denominator is 0.80 × 0.02 + 0.10 × 0.98 = 0.114. The numerator is 0.016. Their ratio is 0.14035.
Expected value is a probability-weighted average
A hypothetical investment gains ₹100 with probability 0.60 and loses ₹50 with probability 0.40.
Neither possible payoff equals ₹40. Expectation averages outcomes across repeated comparable situations. It does not describe the worst case or variability.
Excel illustration
Format the result as a percentage. For a list of outcomes and probabilities, use =SUMPRODUCT(A2:A5,B2:B5). First check that probabilities sum to 1.
Check your understanding
If P(A) = 40%, P(B) = 30% and their overlap is 10%, what is P(A or B)?
40% + 30% − 10% = 60%.
Can a small probability still matter?
Yes. A low probability combined with a very large loss can create a material exposure. Probability and loss severity answer different questions.
Sources and further reading
- National Institute of Standards and Technology: Gallery of Distributions
- Microsoft: Statistical functions reference
All numerical finance and project examples are hypothetical teaching illustrations. Probability assumptions require validation before practical use.