Conditional dependence

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

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

Example

In essence probability is influenced by a person's information about the possible occurrence of an event. For example, let the event ${\displaystyle A}$ be 'I have a new phone'; event ${\displaystyle B}$ be 'I have a new watch'; and event ${\displaystyle C}$ 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 ${\displaystyle C}$ 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 ${\displaystyle \mathrm{\Omega }=\left\{s_1,s_2,s_3,s_4\right\},}$ given in the middle four columns of the following table, in which the occurrence of event ${\displaystyle A}$ is signified by a ${\displaystyle 1}$ in row ${\displaystyle A}$ and its non-occurrence is signified by a ${\displaystyle 0,}$ and likewise for ${\displaystyle B}$ and ${\displaystyle C.}$ That is, ${\displaystyle A=\left\{s_2,s_4\right\},B=\left\{s_3,s_4\right\},}$ and ${\displaystyle C=\left\{s_2,s_3,s_4\right\}.}$ The probability of ${\displaystyle s_i}$ is ${\displaystyle 1/4}$ for every ${\displaystyle i.}$

and so

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