A coin is tossed thrice. In which of the following cases are the events E and F independent?
E : “the number of heads is two”.
F : “the last throw results in head”.

Here,   S = {HHH,  HHT,  HTH,  THH, TTH, THT,  HTT, TTT}
   
     

Here,   S = {HHH,  HHT,  HTH,  THH, TTH, THT,  HTT, TTT}
        P(E) = P(HHH,   TTH, THT, TTH) = 
        
         


   
     

             S = {HHH,  HHT, HTH, THH, HTT, THT, TTH, TTT}
       E = {HHH, TTT},   F = {HHH, HHT, HTH, THH}
and       G = {HHT, HTH,  THH, HTT, THT, TTH, TTT}
       



and  

Also,  

and    

    
             
and     
∴ the events (E and F) are independent, and the events (E and G) and (F and G) are dependent.

Since E and F are independent.
∴  P(E ∩ F) = P(E). P(F)
From the Venn diagram in the figure, it is clear that E ∩ F and E ∩ F' are mutually exclusive events and also
           

∴   E and F' are independent.

Tips: -

Note: In a similar manner, it can be shown that if the events E and F are independent, then
(a)    E' and F are independent,
(b)    E' and F' are independent.

Since A and B are independent events
∴ P(A ∩ B) = P(A) P(B)    ...(1)
P(at least one of A and B) = P(A ∪ B)
= P(A) + P(B) - P(A ∩ B)
= P( A) + P(B) - P(A) P(B)    [∵ of(1)]
= P(A) + P(B) [1 - P(A)]
= P(A) + P(B). P(A') = 1 - P(A') + P(B) P(A')
= 1 - P(A') [1 - P(B)] = 1 - P(A') P(B')