Question

The Probability that atleast one of the two
5
Independent events occur is 0.5. Probability that
first event occurs but not the second is (3/25).
Also the probability that the second event occurs
but not the first is (8/25). Find the probability
that none of the two event occurs

Answer 1

let A and B be the independent events.  They are not exclusive.  Their intersection need not be a null set or null event.    ~A  and ~B are the events when A and B do not occur respectively.

probability that at least one of A or B occur = P (A  U  B) =  1/2

Probability that none of the two events occurs =    P(~A)   AND   P(~B) 
       =  1  -  P(A U B)
        = 1 -  1/2     = 1/2

It is simple.  when none of the events occurs,  it is the compliment of at least one of the events occurs.

===============================
   P (A - B ) = Probability that A occurs but not B
               = P (A) - P(A Π B)
               = probability of A  -  probability of intersection of A and B
  P(A) - P(A Π B) = 3/25

  similarly,  P(B)  -  P(A Π B) = 8/25

we know that
       P(A U B)  =  P (A)  +  [  P(B)  -  P(A Π B) ]
    =>    1/2 = P (A)  + 8/25
    =>  P(A) = 1/2 - 8/25 = 9/25

  Similarly,   P(A U B)  = P(B)  +  [ P(A) - P(A Π B) ]
           =>            1/2  = P(B)  + 3/25
          =>        P(B)  = 1/2 - 3/25 = 19/25

=>  P(A) + P(B) = 28/25
=>  P( A Π B)  =  P (A ) + P(B) - P(A U B)  =  9/25 + 19/25 - 1/2
                     = 31/50
=>  P(~A) = 1 - P(A) = 1 - 9/25 = 16/25
 =>  P(~B) = 1 - P(B) = 1 - 19/25 = 6/25