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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

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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