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.