Question

Probability of happening an event A is $0.5$and that of B is $0.3$. If A and B are mutually exclusive events, then the probability of happening neither A nor B is
A. $0.6$
B. $0.2$
C. $0.21$
D. None of these

Answer 1

The probability of two events, A and B, are given below. We wanted to know the chances of neither A nor B occurring. In probability, neither A nor B have a standard formula. We acquire the answer by substituting values from that standard formula.

Formula used:
Some formulas that we need to know to solve this problem:
$P(A' \cap B') = 1 - P(A \cup B)$
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Complete step by step answer:
It is given that the probability of two events A and B are$0.5$ $\&$ $0.3$respectively and also these events are mutually exclusive.
We aim to find the probability of happening neither A nor B.
We know that there are several standard formulas to find the probability of events. We also have a formula for the probability of neither an event nor the other.
If there are two events A and B. Then, the probability of neither A nor B is given by
$P(A' \cap B') = 1 - P(A \cup B)$
But we don’t know the probability of A union B. Let us find this by using the formula
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Since we already know the values of probability of A and probability of B, substitute them in the above formula.
$P(A \cup B) = 0.5 + 0.3 - P(A \cap B)$
Since it is given that events A and B are mutually exclusive events, we get
$P(A \cup B) = 0.5 + 0.3 - 0$
On simplifying this we get
$P(A \cup B) = 0.8$
Now we got the value of the probability of A union B$P(A \cup B)$. Let us substitute it in the formula$P(A' \cap B') = 1 - P(A \cup B)$.
$P(A' \cap B') = 1 - 0.8$
On simplifying this we get
$P(A' \cap B') = 0.2$
Thus, we got the probability of happening neither A nor B is $0.2$
Let us see the options, option (1) $0.6$is not the correct answer since we got that $0.2$from our calculation.
Option (2) $0.2$is the correct answer as we got the same value in our calculation.
Option (3) $0.21$is not the correct answer since we got that $0.2$from our calculation.
Option (4) None of these is the incorrect answer as we got option (2) as a correct answer.
Hence, option (2) $0.2$is the correct option.

Note: Mutually exclusive events imply that the activities are discontinuous, that there will be no intersection between them. As a result, in the formula above, we substituted 0 for the chance of A intersection B.