Question

If $ P\left( A \right) = 0.4 $ , $ P\left( B \right) = x $ , $ P\left( {A \cup B} \right) = 0.7 $ and the events A and B are two mutually exclusive events, then the value of x is:
(A) $ \dfrac{3}{{10}} $
(B) $ \dfrac{1}{2} $
(C) $ \dfrac{2}{5} $
(D) $ \dfrac{1}{5} $

Answer 1

Hint: Here, $ P(x) $ denotes the probability of some event. Thus, $ P(A) $ means the probability of event A. 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 \cup B} \right) $ means the probability of $ A \cup B $ , that is either of the two events A and B occur. So, we will make use of the formula $ P(A \cup B) = P(A) + P(B) - P(A \cap B) $ to solve the problem and find the value of x.

Complete step-by-step answer:
In the questions, we are given that probability of event A is $ P\left( A \right) = 0.4 $ . Also, Probability that either of the two events A and B occur is $ P\left( {A \cup B} \right) = 0.7 $ .
Now, we are given the probability of event B as variable x. So, we have to find the value of x using the formula $ P(A \cup B) = P(A) + P(B) - P(A \cap B) $ .
We are also provided with the fact 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 \cup B) = P(A) + P(B) - P(A \cap B) $ to solve the problem and find the value of x.
So, we get, $ P(A \cup B) = P(A) + P(B) - P(A \cap B) $
We know that the probability of the two events A and B happening together is zero. So, we have, $ P(A \cap B) = 0 $ . Substituting this into the formula, we get,
 $ \Rightarrow P(A \cup B) = P(A) + P(B) - 0 $
Substituting the values of $ P(A) $ and $ P\left( {A \cup B} \right) $ , we get,
 $ \Rightarrow 0.7 = 0.4 + P(B) $
We also know that $ P\left( B \right) = x $ . So, we get,
 $ \Rightarrow x = 0.7 - 0.4 $
Simplifying the calculations, we get,
 $ \Rightarrow x = 0.3 $
So, we get the value of x as $ 0.3 $ .
We can also write $ 0.3 $ in fractional form as $ \dfrac{3}{{10}} $ .
So, option (A) is the correct answer.
So, the correct answer is “Option A”.

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 \cup B) = P(A) + P(B) - P(A \cap B) $ 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.