Independence of events

Historically, the notion of independence has played a prominent role in probability. If events form an independent class, much lessinformation is required to determine probabilities of Boolean combinations and calculations are correspondingly easier. In this unit, we give a preciseformulation of the concept of independence in the probability sense. As in the case of all concepts which attempt to incorporate intuitive notions, the consequencesmust be evaluated for evidence that these ideas have been captured successfully.

Independence as lack of conditioning

There are many situations in which we have an “operational independence.”

• Supose a deck of playing cards is shuffled and a card is selected at random then replaced with reshuffling. A second card picked on a repeated try should not beaffected by the first choice.
• If customers come into a well stocked shop at different times, each unaware of the choice made by the others, the the item purchased by one should not be affected bythe choice made by the other.
• If two students are taking exams in different courses, the grade one makes should not affect the grade made by the other.

The list of examples could be extended indefinitely. In each case, we should expect to model the events as independent in some way. How should we incorporate the concept in ourdeveloping model of probability?

We take our clue from the examples above. Pairs of events are considered. The “operational independence” described indicates that knowledge that one of the events has occured does notaffect the likelihood that the other will occur. For a pair of events $\{A,B\}$ , this is the condition

Occurrence of the event A is not “conditioned by” occurrence of the event B . Our basic interpretation is that $P\left(A\right)$ indicates of the likelihood of the occurrence of event A . The development of conditional probability in the module Conditional Probability , leads to the interpretation of $P\left(A\right|B)$ as the likelihood that A will occur on a trial, given knowledge that B has occurred. If such knowledge of the occurrence of B does not affect the likelihood of the occurrence of A , we should be inclined to think of the events A and B as being independent in a probability sense.