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.