Addition Rule for Probabilities

What Is the Addition Rule for Probabilities?

The addition rule for probabilities calculates the probability that event A or event B occurs. It is used whenever the question is about at least one event happening rather than both events happening together.

Events can overlap. If overlap exists, simply adding the two probabilities double-counts the cases where both events happen. In finance, that mistake can distort risk estimates when several bad outcomes can occur in the same scenario.

Formula

For two events, the general addition rule is:

P(A) is the probability of event A. P(B) is the probability of event B. P(A and B) is the probability that both events occur. Subtracting the overlap keeps the same combined scenario from being counted twice.

If A and B are mutually exclusive, then P(A and B) is zero, so the rule becomes P(A or B) = P(A) + P(B).

Credit-Risk Example

Suppose there is a 20% chance a borrower misses a payment this year and a 10% chance the borrower breaches another loan covenant. If there is a 4% chance both happen, the probability of at least one of the two problems is 26%, not 30%.

The overlap matters because the same bad year may cause both events. If the borrower loses a major customer, that single stress event could make both missed payments and covenant breaches more likely.

Mutually Exclusive Versus Overlapping Events

Events are mutually exclusive when they cannot occur together. A bond cannot both default and not default over the same defined period. In that kind of setup, adding probabilities can be straightforward as long as the categories are truly separate.

Many financial events are not mutually exclusive. A recession, a market drawdown, a credit downgrade, a liquidity squeeze, and a dividend cut can cluster together. The addition rule forces the analyst to ask whether events are separate, overlapping, or driven by the same underlying stress.

Risk Analysis Use

Risk work often combines scenarios. If analysts add event probabilities without checking overlap, they can overstate total risk. If they assume events are unrelated when they are actually connected, they can understate stress-case clustering.

The addition rule does not solve every probability problem. It simply keeps the basic counting clean. Good risk analysis still needs realistic assumptions about correlation, time period, data quality, and whether the probabilities came from history, models, judgment, or market prices.

The Bottom Line

The addition rule for probabilities is a guardrail against double-counting. It is especially useful when financial risks overlap rather than arriving as clean, isolated events.