Math
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# Independence

Lesson 8 Chapter 2

We previously discussed conditional probabilities. You learn the concept of probability of an event dependent to another event as . Intuitive, we can infer that if and are independent, then does not depend on , and it simply equals to . We previously investigated the formulation as well. Technically, being independent is mutual, i.e., when , then and vice verse. We can conclude that when and are independent, then . This formulation leads to the following definition:

### Independent Events

Two events and are said to be independent if . Otherwise, they are called dependent.

### Example

Assume we toss two dices. Let denote the event that the sum of the dice is 5 and indicate the event that the first dice equals 3. Can you confirm if the two events are independent or not?

Solution: To find the answer, we should calculate and . If both are equal, then the two events are independent. Otherwise, they are dependent. How we did the calculations? Calculating is simple for only one dice, the probability of getting any number is . But how are we going to calculate ? The event space of is the following: Henceforth, equals to . Basically, represents all the possible situations that the sum of two dices equals to five. Since , then we can conclude that and are NOT independent.