Question

The probability that at least one of the event A and B occurs is 0.6, If A and B occur simultaneously with probability 0.2, then $P\left( {\bar A} \right) + P\left( {\bar B} \right)$ is
(a) 0.4
(b) 0.8
(c) 1.2
(d) 1.4

Answer 1

Hint – In this question at least one event occurs means $P\left( {A \cup B} \right)$ and two events occur simultaneously means $P\left( {A \cap B} \right)$. Use the concept that probability of addition up of an event and non-occurrence of one event is always equal to one that is $P\left( A \right) + P\left( {\bar A} \right) = 1$. This will help getting the answer.

Complete step-by-step answer:
Given data:
Probability that at least one of the events A and B occurs is 0.6.
$ \Rightarrow P\left( {A \cup B} \right) = 0.6$
And if A and B occur simultaneously the probability is 0.2.
$ \Rightarrow P\left( {A \cap B} \right) = 0.2$
So we have to find out the value of $P\left( {\bar A} \right) + P\left( {\bar B} \right)$, where $P\left( {\bar A} \right)$ and $P\left( {\bar B} \right)$ is the probability of not occurring the events A and B.
As we know that the total probability is 1 (i.e. the sum of probability of occurring and the probability of not occurring).
$ \Rightarrow P\left( A \right) + P\left( {\bar A} \right) = 1$
$ \Rightarrow P\left( A \right) = 1 - P\left( {\bar A} \right)$.................. (1)
Similarly,
$ \Rightarrow P\left( B \right) = 1 - P\left( {\bar B} \right)$................ (2)
Now according to set relation we have,
$ \Rightarrow P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)$
Now from equation (1) and (2) we have,
$ \Rightarrow P\left( {A \cup B} \right) = 1 - P\left( {\bar A} \right) + 1 - P\left( {\bar B} \right) - P\left( {A \cap B} \right)$
$ \Rightarrow P\left( {\bar A} \right) + P\left( {\bar B} \right) = 2 - P\left( {A \cup B} \right) - P\left( {A \cap B} \right)$
Now substitute the values we have,
$ \Rightarrow P\left( {\bar A} \right) + P\left( {\bar B} \right) = 2 - 0.6 - 0.2 = 2 - 0.8 = 1.2$
So this is the required answer.
Hence option (C) is the correct answer.

Note – $P\left( {\bar A} \right)$ means the probability of non-occurrence of event A. So $P\left( {\bar A} \right) + P\left( {\bar B} \right)$ means that we were to find the probability of non-occurrence of two events A and B. The probability is always in between 0 to 1 that means $0 \leqslant P(A) \leqslant 1$, thus the compliment or non-occurrence of any event will simply be $1 - P(A)$.