Conditional Probability – Independent Events

This chapter highlights the importance of identifying the correct sample space when the original space is limited due to conditioning upon some event. In many real life problems, families of independent events are put in some order such as logical and chronological. If such an ordering is possible, it is often easy to calculate conditional probabilities directly, upon conditioning each time on the events that are observed. A very common problem in probability theory is the calculation of the a posteriori probabilities based on the a priori probabilities and the conditional probabilities. The general expression for that calculation is in fact an application of the multiplicative law and the law of total probability and is attributed to Reverend Thomas Bayes. Bayes' theorem has found a large variety of applications in statistics and for this reason it has attracted a lot of interest among statisticians.