Independence and Dependence of Events

1. Independence : Two events and are said to be independent if the occurrence of event does not affect the probability of event occurring. That is,
2. Dependence : Two events and are dependent if the occurrence or non-occurrence of event affects the probability of event . That is,
3. Multiplication Rule for Independent Events
   The necessary and sufficient condition for two events and to be independent is: provided that
   
   
4. Comparison Between Mutually Exclusive Events and Independent Events

   	Mutually Exclusive Events	Independent Events
   Definition
   Meaning	Cannot occur simultaneously	Do not affect each other
   Addition and Multiplication Rules
   How to Determine	If , then and are mutually exclusive	If , then and are independent

   For two events and , where :
1. If and are mutually exclusive, then they are not independent.
2. If and are independent, then they are not mutually exclusive.

 

Example. Given that two events and are independent and satisfy the following conditions:

,
Find .

 Solution

Since events and are independent, the two events and are also independent.

From the equation ,

By substituting the given values,

Therefore,

Probability of Independent Trials

1. Independent Trials : When an experiment is repeated, and the outcome of each trial is independent of the others, these trials are called independent trials.
2. Probability of Independent Trials
   If the probability of event occurring in one trial is , then the probability that event occurs times in independent trials is given by:

 

Example. Vera and Everett participated in a marathon. The probability that at least one of them will finish is , and the probability that Vera will finish is . Find the probability that Everett will finish. (Assume that the events of Vera and Everett finishing are independent of each other.)

 Solution

Let the events that Vera and Everett finish the marathon be denoted as A and B, respectively.

,
Since the events A and B are independent,

Using the formula for the union of two events,

Substitute the given values and solve for

,
Therefore the probability that Everett will finish is