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Conditional dependence

Concept in probability theory

Wikipedia contributorsAdapted for Fathomly · View authors
2 min read
Source updated November 18, 2025
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In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs. It is the opposite of conditional independence. For example, if A and B are two events that individually increase the probability of a third event C, and do not directly affect each other, then initially (when it has not been observed whether or not the event C occurs) \operatorname {P} (A\mid B)=\operatorname {P} (A)\quad {\text{ and }}\quad \operatorname {P} (B\mid A)=\operatorname {P} (B) (A{\text{ and }}B are independent).

But suppose that now C is observed to occur. If event B occurs then the probability of occurrence of the event A will decrease because its positive relation to C is less necessary as an explanation for the occurrence of C (similarly, event A occurring will decrease the probability of occurrence of B). Hence, now the two events A and B are conditionally negatively dependent on each other because the probability of occurrence of each is negatively dependent on whether the other occurs. We have \operatorname {P} (A\mid C{\text{ and }}B)<\operatorname {P} (A\mid C).

Conditional dependence of A and B given C is the logical negation of conditional independence ((A\perp \!\!\!\perp B)\mid C). In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event.

01Example

In essence probability is influenced by a person's information about the possible occurrence of an event. For example, let the event A be 'I have a new phone'; event B be 'I have a new watch'; and event C be 'I am happy'; and suppose that having either a new phone or a new watch increases the probability of my being happy. Let us assume that the event C has occurred, meaning 'I am happy'. Now if another person sees my new watch, he/she will reason that my likelihood of being happy was increased by my new watch, so there is less need to attribute my happiness to a new phone.

To make the example more numerically specific, suppose that there are four possible states \Omega =\left\{s_{1},s_{2},s_{3},s_{4}\right\}, given in the middle four columns of the following table, in which the occurrence of event A is signified by a 1 in row A and its non-occurrence is signified by a 0, and likewise for B and C. That is, A=\left\{s_{2},s_{4}\right\},B=\left\{s_{3},s_{4}\right\}, and C=\left\{s_{2},s_{3},s_{4}\right\}. The probability of s_{i} is 1/4 for every i.

Event\operatorname {P} (s_{1})=1/4\operatorname {P} (s_{2})=1/4\operatorname {P} (s_{3})=1/4\operatorname {P} (s_{4})=1/4Probability of event
A0101 {\tfrac {1}{2}}
B0011 {\tfrac {1}{2}}
C0111 {\tfrac {3}{4}}

and so

Events_{1}s_{2}s_{3}s_{4}Probability of event
A\cap B0001 {\tfrac {1}{4}}
A\cap C0101 {\tfrac {1}{2}}
B\cap C0011 {\tfrac {1}{2}}
A\cap B\cap C0001 {\tfrac {1}{4}}

In this example, C occurs if and only if at least one of A,B occurs. Unconditionally (that is, without reference to C), A and B are independent of each other because \operatorname {P} (A), the sum of the probabilities associated with a 1 in row A, is {\tfrac {1}{2}}, while \operatorname {P} (A\mid B)=\operatorname {P} (A{\text{ and }}B)/\operatorname {P} (B)={\tfrac {1/4}{1/2}}={\tfrac {1}{2}}=\operatorname {P} (A). But conditional on C having occurred (the last three columns in the table), we have \operatorname {P} (A\mid C)=\operatorname {P} (A{\text{ and }}C)/\operatorname {P} (C)={\tfrac {1/2}{3/4}}={\tfrac {2}{3}} while \operatorname {P} (A\mid C{\text{ and }}B)=\operatorname {P} (A{\text{ and }}C{\text{ and }}B)/\operatorname {P} (C{\text{ and }}B)={\tfrac {1/4}{1/2}}={\tfrac {1}{2}}<\operatorname {P} (A\mid C). Since in the presence of C the probability of A is affected by the presence or absence of B,A and B are mutually dependent conditional on C.

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Sources and credits

This article is adapted from the Wikipedia article “Conditional dependence”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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