What it means
Ask how likely it is that either of two things happens, and the intuitive answer is to add the two probabilities. The addition rule tells you when that works and when it quietly double-counts.
The rule has two forms. For mutually exclusive events (outcomes that cannot occur together), the probability of either is simply the sum.
Rolling a die shows the simple case: the chance of a 3 or a 6 is 1/6 plus 1/6, which is 1/3, because a single roll cannot be both. The second form handles events that can overlap.
The probability of Y or Z equals P(Y) plus P(Z) minus P(Y and Z), where the last term removes the double-counted intersection. The first form is really the second with an overlap of zero, so one formula covers both cases.
A classroom makes it concrete. In a class of 9 boys and 11 girls, 5 girls and 4 boys earned a B.
The chance that a randomly picked student is a girl or a B student is 11/20 plus 9/20 minus 5/20, which is 15/20 or 75%, because the 5 girls with a B were counted twice before the subtraction. In business, this is the workhorse of everyday risk arithmetic.
The chance a shipment is late or damaged, the chance a project misses its date or its budget, and the chance either of two machines fails today are all addition-rule problems. The overlap term often comes from the multiplication rule, which gives the probability that both events happen.
The common error is adding without checking the overlap. If two risks can strike together, plain addition overstates the combined probability and can even produce answers above 100%.
A manager's discipline is three questions: what are the individual probabilities, can the events happen together, and if so, how big is the overlap?
In practice
Real-world examples.
Example
A logistics manager estimates a 10% chance a shipment is delayed by weather and a 5% chance it is delayed by port congestion, with a 2% chance of both. The chance of a weather-or-congestion delay is 10% + 5% - 2% = 13%.
Example
A quality engineer models two independent defect modes in a batch, each with a 4% chance of occurring. Because they are independent, the chance of both is 4% x 4% = 0.16%, so the chance of at least one defect mode is 4% + 4% - 0.16% = 7.84%.
Example
A retailer finds that 30% of customers open its promotional email and 20% see its app notification, while 8% do both. The share reached by at least one channel is 30% + 20% - 8% = 42%.
Formula
Calculation
Mutually exclusive events: P(Y or Z) = P(Y) + P(Z). Overlapping events: P(Y or Z) = P(Y) + P(Z) - P(Y and Z).
Worked example. A class has 20 students: 11 girls and 9 boys. Nine students earned a B, made up of 5 girls and 4 boys. P(girl) = 11/20 = 55%, P(B) = 9/20 = 45%, and P(girl and B) = 5/20 = 25%. So P(girl or B) = 55% + 45% - 25% = 75%. As a check, count directly: all 11 girls plus the 4 boys with a B gives 15 students, and 15/20 = 75%.Case study
Seen in the real world.
Brightwell Civil is an invented construction firm used here for illustration. Its project sponsor adds the 20% chance of a supplier delay to the 15% chance of a permit delay and reports a 35% schedule risk to the board. The number looks alarming but is not quite right.
In this fictional story a risk analyst points out that both delays share a common cause in bad weather, and estimates a 6% chance that both occur. The corrected figure is 20% + 15% - 6% = 29%, and that smaller number changes the decision, because it falls below the threshold at which the board requires a contingency buffer.
Watch out
Common mistakes.
- Adding probabilities without checking for overlap, because when events can occur together the sum counts the intersection twice and overstates the risk.
- Assuming events are mutually exclusive when they are not, for example when one cause such as bad weather can hit both shipping and staffing.
- Confusing mutually exclusive with independent. Exclusive events cannot happen together, so knowing one occurred tells you the other did not, which makes them strongly dependent.
Questions
People also ask.
What is the addition rule for probabilities?
It is the formula for the chance that at least one of two events occurs: add their probabilities, and if the events can happen together, subtract the probability of both to remove the double count.
What are mutually exclusive events?
They are events that cannot happen at the same time, such as rolling a 3 and a 6 on one die. Their overlap is zero, so their probabilities simply add.
How does the addition rule relate to the multiplication rule?
The multiplication rule gives the probability that both events occur, which is exactly the overlap term the addition rule subtracts. The two rules often work as a pair.
From the founder's library
Take it further with the book.
Build your financial confidence beyond this definition. Shihan's full-length guide, Accounting Fundamentals, takes the same plain-English approach and turns it into a complete, practical playbook for non-finance managers, business owners and students - with chapter-end quiz answers and presentation slides included.
25% off with code MMHQ25, applied at checkout. Priced in USD - checkout may show the equivalent in your local currency.
View the book and save 25%Related