Two events are said to independent if the occurance of one event does not affect the occurence of the other event. This implies that the probability of one event will not change even after it is known that other event is already happened. Mathematically, \(P(F|E) = P(F)\)
Or when we expand P(F | E) using conditional probabaility, we get
\(P(F|E) = P(E\cap F)\div P(E)\)
Which is further equal to P(F). Note that,f or P(E)=0, P(F|E) cannot be defined. Equating both of these equations, we get the condition to determine whether two events are independent or not. i.e.
\(P(E\cap F) = P(E).P(F)\)
Before we get into further discussion about why this is this important to identify whether the events are independent or not, note that, the above condition can be generalized over N events.
When you deal with real world data, you'll hardly find independent events. For example, if you are building a model that predict whether there will be raining on a certain day or not, you'll have features like clouds, humidity, temperature, etc., and each of these feature will affect your probability of raining. These are independent events. But, if you have some independent features like amount of rainfall on the same day last year, Ii is very unlikely that they'll affect the probability for of rainfall. Thus, it becomes extremely important to identify the independent events (features) during feature selection. The creadibility of your final model will depend on it.
You can easily check for indepent features in Python using the above mentioned condition. These can be applied in the case of sets too. This can also be applied to real world games like dice, cards, or Roulette.
This was just an introduction about dependent and independent events. The concepts we covered here can be applied to much complex cases like determining the risk of surgery if the person is diabetic or determining the probability of winning the world cup if last season's champion is eliminated. You need to have a clear understanding of independence in order to apply this in real world scenarios.