Conditional Probability, Uses

Conditional probability measures the likelihood of an event $A$ occurring, given that another event $B$ has already occurred.

Many events associated to the real experiments, specifically related to civil engineering, have some dependency on the other events associated to the same experiment. In that case the occurrence of the dependent event is a function of the occurrence of the other events. One such example is given the probability of having cube density in between two values what is the probability of having the cube strength ranging in between two specific values.

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  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.

Uses of Conditional Probability:

• Medical Diagnostics

Conditional probability is extensively used in healthcare to predict the likelihood of a disease given specific symptoms or test results. For example, the probability of a patient having a disease (event $A$) given a positive test result (event $B$) helps in assessing the effectiveness of medical tests using concepts like sensitivity and specificity.

• Weather Forecasting

Meteorologists use conditional probability to estimate the likelihood of weather events. For instance, the probability of rain ($A$) given specific atmospheric conditions ($B$) like humidity or pressure. This helps in making informed decisions regarding agriculture, travel, and outdoor activities.

• Risk Assessment in Finance

In financial markets, conditional probability is employed to evaluate the likelihood of market downturns or defaults given current economic indicators or historical data. For example, the probability of a stock’s value decreasing ($A$) given an economic slowdown ($B$) aids in portfolio management and decision-making.

• Machine Learning and AI

Conditional probability forms the foundation of probabilistic models like Bayesian Networks and Naive Bayes classifiers. These models predict outcomes based on the likelihood of related events, such as spam detection (A) given email attributes (B) or customer purchasing behavior based on preferences.

• Reliability Engineering

In engineering, conditional probability is used to calculate the reliability of systems. For example, the probability of a machine part failing ($A$) given that another part has failed ($B$) helps in maintenance scheduling and designing fail-safe systems to ensure operational continuity.