What it means
Every business plan is a chain of events, such as winning a tender, hiring a team, shipping on time and being paid. Each link has its own probability, and compound probability combines them.
The combined figure is almost always lower than the weakest individual link when all steps must succeed. There are two common cases.
When all events must happen (A and B), and the events do not influence each other, you multiply the probabilities. When you only need at least one event to happen (A or B), you add the probabilities and subtract the overlap, or more simply calculate the chance that none happen and take that from 100%.
The word "independent" matters a great deal. Independent events do not affect one another, like two unrelated customers each deciding whether to renew.
Dependent events do affect one another, so the second probability must be adjusted to reflect that the first has already happened. Finance teams use compound probability in risk registers, scenario analysis, credit modelling and forecast reviews.
It helps answer questions such as how likely it is that three suppliers all deliver late, or that a project hits budget, schedule and quality targets together. The practical lesson is a warning about over-optimism.
Five steps that are each 90% likely look safe, yet together they succeed only about 59% of the time. Managers who ignore this often approve plans that are far riskier than they appear.
In practice
Real-world examples.
Example
A manufacturer needs two independent suppliers to deliver on time for a product launch, and each has a 95% on-time record. The chance both deliver on time is 0.95 x 0.95 = 0.9025, or about 90.3%. The launch is therefore at risk almost 10% of the time even though each supplier looks reliable.
Example
A sales director has three unrelated prospects, each with a 20% chance of signing this quarter. The chance none sign is 0.80 x 0.80 x 0.80 = 0.512, so the chance at least one signs is 48.8%. She uses that figure to set a realistic quarterly target.
Example
A clinic chain tests a new booking system in two cities, where success in each is judged separately at 60%. The chance both pilots succeed is 0.60 x 0.60 = 0.36, or 36%. The board decides to stage the rollout rather than commit the full budget at once.
Formula
Calculation
For independent events: P(A and B) = P(A) x P(B). For at least one event: P(A or B) = 1 - [(1 - P(A)) x (1 - P(B))].
Suppose a start-up needs three things to happen to close a $400,000 contract: the pilot succeeds (80%), the budget is approved (70%) and the legal review passes (90%). Assume the three events are independent.
P(all three) = 0.80 x 0.70 x 0.90 = 0.56 x 0.90 = 0.504, which is 50.4%.
Expected value of the contract = 0.504 x $400,000 = $201,600. Now suppose the team has two separate leads that each have a 30% chance of closing. The chance that at least one closes is 1 - (0.70 x 0.70) = 1 - 0.49 = 0.51, or 51%.Case study
Seen in the real world.
Northfield Logistics is a fictional courier company used for illustration. Its leadership approved a $2,000,000 depot expansion after reviewing four milestones, each rated 85% likely: planning consent, contractor delivery, staff recruitment and a key customer contract.
An analyst pointed out that the plan needed all four to succeed. Multiplying 0.85 x 0.85 x 0.85 x 0.85 gives roughly 0.522, so the plan had only about a 52% chance of working as designed. The board responded by splitting the investment into two phases, securing the customer contract first and releasing capital only once that milestone was met.
Watch out
Common mistakes.
- Assuming the overall chance equals the average of the individual chances. When all steps must succeed, the combined probability is the product and is lower than any single step.
- Multiplying probabilities for events that are linked. If one failure makes another more likely, the independence assumption breaks and the answer is too optimistic.
- Adding probabilities for "and" situations. Adding is only part of the method for "or" situations, and it requires subtracting the overlap.
Questions
People also ask.
What is the difference between compound and conditional probability?
Compound probability covers several events together, while conditional probability is the chance of one event given that another has already happened. Conditional values are often the inputs for dependent compound calculations.
How do I know if events are independent?
Ask whether knowing the outcome of one would change your view of the other. If it would, treat them as dependent.
Can compound probability be used with forecasts?
Yes, it is useful for testing whether a forecast depends on too many things going right. Even good individual estimates can combine into a modest overall chance.
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