What is the difference between independent events and dependent events?

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Independent Events The probability of one event happening isn't influenced by the outcome of another.

In mathematical terms

# A,B " events", P(AnnB)=P(A)*P(B)#
This reads: Probability of A and B happening is equal to the probability of A happening multiplied by the probability of B happening

An equivalent definition

#P(A|B) = P(A)#

This reads: The probability of A given B happened is equal the probability of A

Example:
Event A: Rolling a number larger than four on a die in the first roll Event B: Rolling a 6 on a die on the second roll. These events are independent because one roll of a dice doesn't influence the outcome of another dice roll.

Dependent Events Is the oposite.If the outcome of one event influences the probability of the other they aren't independent

#P(A|B) != P(A)#

Example:
Event A: Rolling a number larger than four on a die Event B: Rolling a 6 on a die on the same roll. These events are dependent because ,as we are in the same dice roll, the occurrence of event B changes the probability of event A, actually it makes it 1.

Observation:
Pay attention that if you are dealing with more than 2 event, for them to be mutually independent every possible combination of them must be. For example, if we are dealing with 3 events, independence means that:

#P(AnnBnnC)=P(A)*P(B)*P(C)#
#" and " P(AnnB) = P(A)*P(B), P(BnnC)=P(B)*P(C), P(CnnB)=P(C)*P(B)#

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