Question

If $A$ and $B$ are two mutually exclusive events, then the value of $P\left( {A + B} \right)$ is:
(A) $P\left( A \right) + P\left( B \right) - P\left( {AB} \right)$
(B) $P\left( A \right) - P\left( B \right)$
(C) $P\left( A \right) + P\left( B \right)$
(D) $P\left( A \right) + P\left( B \right) + P\left( {AB} \right)$

Answer 1

Hint : Here, $P(x)$ denotes the probability of some event. Thus, $P(AB)$ means the probability of $AB$ , that is both the events A and B occur. We are given that the events A and B are mutually exclusive events. So, there is nothing common between the two events. Also, $P\left( {A + B} \right)$ means the probability of $A + B$ , that is either of the two events A and B occur. So, we will make use of the formula $P(A + B) = P(A) + P(B) - P(AB)$ to solve the problem and find the value of $P\left( {A + B} \right)$ .

Complete step-by-step answer :
In the given question, we have to find the probability that either of the two events A and B occur given that A and B are two mutually exclusive events. So, there is nothing common in the two events A and B and they cannot happen simultaneously.
Hence, the probability of the two events A and B happening together is zero.
Now, we will use the formula
$P(A + B) = P(A) + P(B) - P(AB)$ to solve the problem and find the value of $P\left( {A + B} \right)$ .
So, we get,
$P(A + B) = P(A) + P(B) - P(AB)$
We know that the probability of the two events A and B happening together is zero. So, we have, $P(AB) = 0$ . Substituting this into the formula, we get,
$\Rightarrow P(A + B) = P(A) + P(B) - 0$
Simplifying the calculations, we get,
$\Rightarrow P(A + B) = P(A) + P(B)$
So, we get the value of $P(A + B)$ as $P(A) + P(B)$ given that the two events A and B are mutually exclusive events.
So, option (C) is the correct answer.
So, the correct answer is “Option C”.

Note : These problems are the combinations of sets and probability, so, the concepts of both of the topics are used in these. Here the formula, $P(A + B) = P(A) + P(B) - P(AB)$ is used. This formula is a restructured version of the formula of sets, which is, $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ where, $n(x)$ denotes the number of elements in set $x.$ This formula is modified into the formula of probability by dividing on both sides by $n(U)$ , where $U$ is the universal set.