In essence probability is influenced by a person's information about the possible occurrence of an event. For example, let the event be 'I have a new phone'; event be 'I have a new watch'; and event 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 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 given in the middle four columns of the following table, in which the occurrence of event is signified by a in row and its non-occurrence is signified by a and likewise for and That is, and The probability of is for every
and so
In this example, occurs if and only if at least one of occurs. Unconditionally (that is, without reference to ), and are independent of each other because —the sum of the probabilities associated with a in row —is while
But conditional on having occurred (the last three columns in the table), we have
while
Since in the presence of the probability of is affected by the presence or absence of and are mutually dependent conditional on