Maharashtra State BoardHSC Science (General) 11th
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Independent Events

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definition

Two events E and F are said to be independent, if 
P(F|E) = P (F)  provided P (E)  ≠ 0 
and P(E|F) = P (E)  provided P (F)  ≠ 0 
Thus, in this definition we need to have 
P (E) ≠ 0 and  P(F) ≠ 0 
Now, by the multiplication rule of probability, we have 
P(E ∩ F) = P(E) . P (F|E)              ... (1) 
If E and F are independent, then (1) becomes 
P(E ∩ F) = P(E) . P(F)                   ...(2) 

definition

Let E and F be two events associated with the same random experiment, then E and F are said to be independent if 
P(E ∩ F) = P(E) . P (F) 

notes

 If E and F denote the events 'the card drawn is a spade' and 'the card drawn is an ace' respectively, then 
P(E) = `13/52` = `1/4` and P(F) = `4/52 = 1/13`
Also E and F is the event ' the card drawn is the ace of spades' so that 
P(E ∩F) = `1 /52`
Hence  P(E|F) = `(P(E ∩ F))/(P(F)) = 1/52 / 1/13 = 1/4`
Since P(E) = `1/4` = P (E|F), we can say that the occurrence of event F has not affected the probability of occurrence of the event E. 
We also have P(F|E) = `(P( F ∩ E))/(P(E)) = 1/52 /1/4 = 1/13 = P(F)`
Again,  P(F) =  `1/13` =  P(F|E)  shows that occurrence of event E has not affected  the probability of occurrence of the event F. Thus, E and F are two events such that the probability of occurrence of one of them is not affected by occurrence of the other. Such events are called independent events. 

Remarks :

(i) Two events E and F are said to be dependent if they are not independent, i.e. if  
P(E ∩ F ) ≠ P(E) . P (F) 

ii) Sometimes there is a confusion between independent events and mutually exclusive events. Term ‘independent’ is defined in terms of ‘probability of events’ whereas mutually exclusive is defined in term of events (subset of sample space). Moreover, mutually exclusive events never have an outcome common, but independent events, may have common outcome. Clearly, ‘independent’ and ‘mutually exclusive’ do not have the same meaning. 
In other words, two independent events having nonzero probabilities of occurrence can not be mutually exclusive, and conversely, i.e. two mutually exclusive events having nonzero probabilities of occurrence can not be independent. 

(iii) Two experiments are said to be independent if for every pair of events E and F, where E is associated with the first experiment and F with the second experiment, the probability of the simultaneous occurrence of the events E and F when the two experiments are performed is the product of P(E) and P(F) calculated separately on the basis of two experiments, i.e., 
P (E ∩ F) = P (E) . P(F)
(iv) Three events A, B and C are said to be mutually independent, if 
P(A ∩ B) = P(A) P(B) 
P(A ∩ C) = P(A) P(C) 
P(B ∩ C) = P(B) P(C) 
and P(A ∩ B ∩ C) = P(A) P(B) P(C) 
Video link : https://youtu.be/vAnUlZdgZXE

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