Dependent Events

When events A and B are not independent, they are dependent. This means that the occurrence of event A affects the probability of event B happening. This often means that event A occurs first, and the probability of event B changes because event A has already happened but not always.
Thunder and lightning occur together. Lightning occurs first, and lightning produces the thunder that we hear. Thunder and lightning are dependent events. The thunder is dependent on the lightning occurring.
Thunder and lightning are correlated with rain, but it can rain before after or during lightning, so even though rain and lightning are dependent, it is not the case that the occurrence of rain is dependent on lightning having occurred or vice versa.
If events A and B are independent we cannot use the equations 
\[P(A \cap B)=P(A) \times P(B)\]
  (the probability of A and B both happening is equal to the probability of A happening times the probability of B hpppening) and  
\[P(A \| B)+P(A)\]
(the probability of A happening given that B has happened equals the probability of A happening).
We have to use the equation  
\[P(A)=\frac{(A \cap B)}{P(B)}\]

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