Can probabilities be multiplied when events are dependent?

Yes, using the appropriate conditional probabilities in the general multiplication rule.

Does the order matter?

Correctly specified conditional forms give the same joint probability, although different orders use different conditional quantities.

What if the first event has probability zero?

Its intersection with another event also has probability zero, but the elementary conditional probability given that event is not defined by division.

AI Glossary / Statistics and probability / Foundation

Multiplicative law of probability

Definition

The multiplicative law of probability, usually called the multiplication rule, expresses the probability that events occur together using a marginal probability and a conditional probability. For events A and B with P(A) greater than zero, P(A and B) = P(A) times P(B given A). Equivalently, when P(B) is positive, it equals P(B) times P(A given B). If A and B are independent, the conditional probability equals the corresponding marginal probability, giving the simpler product P(A)P(B). Independence is a special case, not a requirement for the general rule.

What it means

A joint event requires every stated condition to hold. The multiplication rule handles this by first accounting for one condition and then considering the other within the cases where the first holds. Conditioning changes the reference population. The rule is therefore useful for sequential stages, dependent draws and intersections that have no temporal order. Writing one event first does not imply it causes the other. The key is using probabilities that refer to compatible populations, time periods and conditioning information.

How it works

Define the events clearly and identify the joint event of interest. Choose an order in which relevant conditional probabilities are available. Multiply the first event's probability by the probability of the next event given the first. For three events, use P(A) times P(B given A) times P(C given A and B). Continue this chain for additional events when the required conditioning probabilities are defined. Replace conditional probabilities with marginals only when the corresponding independence assumptions are justified. Check that the resulting intersection probability does not exceed the probability of any individual event.

Three practical examples

Workflow analysis: estimate the probability of completing several stages using stage-success probabilities conditional on reaching the relevant stage.
Sampling: calculate the probability of a specified sequence of draws when earlier draws change what remains.
Reliability reasoning: combine component or stage events while accounting for shared conditions instead of assuming independence. The numerical inputs need evidence, and the rule does not establish that the assumed dependence structure is correct.

Think of it this way

Think of passing through two gates. First determine what fraction of the original group passes the first gate. Then determine what fraction of that smaller group passes the second. Multiplying the two fractions gives the fraction passing both. Using the second gate's rate from a different population can produce the wrong answer even though the multiplication itself is correct.

Worked example

Suppose 60% of applications pass an initial screen, and 75% of those that pass the screen complete the next stage. The probability of both events is 0.60 times 0.75 = 0.45, or 45% of all applications. If 80% of applications completing both stages then finish a third stage, the probability of completing all three is 0.45 times 0.80 = 0.36. The third percentage is conditional on both earlier stages. Separately, draw two tokens without replacement from a bag with three red and two blue tokens. The probability both are red is (3/5) times (2/4) = 0.30. Multiplying (3/5) by itself would give 0.36 and incorrectly assume the second draw's red probability remains unchanged after a red token is removed. These original calculations illustrate conditional multiplication in two settings.

Illustrative case study

In a fictional case, Alder Operations estimates how many requests will complete a multistage process. Different teams provide percentages using different denominators: all requests, screened requests and requests already approved. The analyst maps each percentage to its conditioning group before constructing the calculation. The revised report shows both the stage-specific rates and the implied overall completion rate. It also checks whether the rates come from comparable periods, since combining unrelated operating conditions would not produce a reliable forecast.

Common misconceptions and mistakes

Multiplying marginal probabilities is not generally valid for dependent events. Mutually exclusive events with positive probabilities are not independent: if one occurs, the other cannot. The multiplication rule concerns an intersection, while a union requires an addition rule with appropriate overlap handling. Conditional probability does not imply causation or temporal sequence. Pairwise independence of several events does not always imply mutual independence, so multiplying every marginal in a long chain can require stronger assumptions. Conditioning on an event of zero probability needs more careful mathematical treatment than the elementary ratio formula.

Business use

Businesses should make denominators explicit whenever combining conversion rates, completion rates or risk estimates. Record whether a percentage applies to all cases or only those satisfying earlier conditions. Look for common causes that make events dependent, such as a shared outage affecting several components. Test the resulting model against observed joint outcomes and update assumptions when conditions change. A transparent probability chain can reveal where a process loses cases, but it should not be used to manufacture precision from incompatible or weakly estimated inputs.

Check your understanding

P(A) = 0.6 and P(B given A) = 0.75. What is P(A and B)?

Answer: 0.45. Knowledge of P(B) alone would not replace the conditional value unless independence were justified.

Related terms

Learn next

Practice converting percentage statements into explicit conditional probabilities. Then use probability trees and contingency tables to check multiplication calculations and distinguish joint, marginal and conditional quantities.

Sources

Penn State STAT 414, Conditional Probability and multiplication rule: https://online.stat.psu.edu/stat414/Lesson04 Penn State STAT 414, Independent Events: https://online.stat.psu.edu/stat414/Lesson05

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