Conditional probability refers to the probability of an event occurring, given that another event has already occurred. It quantifies the likelihood of one event under the condition that the related event is known.
The probability of the occurrence of an event A given that an event B has already occurred is called the conditional probability of A given B:
The same is explained in Figure 2.15 using the sample spaces related to the events A and B, assuming that there are few sample points common to these two events. Part 1 of the figure shows the total sample space related to the experiment as in the form of rectangle and the sample space related to the event A as a circle. Similarly part 2 of the figure shows the total sample space and the sample space related to event B. As explained earlier in conditional probability the total sample space is restrained to the sample space that is related to event B (which has already occurred). The same is shown in part 3 of Figure 2.15. Now the sample space for event A (B is the total sample space available) is nothing but the sample points related to event A and falling in the sample space. This is nothing but the intersection of the events A and B and is shown in part 3 of the figure as the hatched area.
Figure 2.15: Representation of conditional probability using the Venn diagrams
For example, there are 100 trips per day between two places X and Y. Out of these 100 trips 50 are made by car, 25 are made by bus and the other 25 are by local train. Probabilities associated to these modes are 0.5, 0.25, and 0.25, respectively. In transportation engineering both the bus and the local train are considered as public transport so the event space associated to this is the summation of the event spaces associated to bus and local train. Probability of choosing public transportation is 0.5. Now if one is interested in finding the probability of choosing bus given public transportation is chosen the conditional probability is useful in finding that.