Question

If $A$ and \[B\] are two events such that \[P(A)=0.4\] \[P(A+B)=0.7\], \[P(A\cap B)=0.2\] and, $P(B)=?$
A. $0.1$
B. $0.3$
C. $0.5$
D. None of these

Answer 1

Hint: In this question, we have to find probability of event \[B\]. In this question, the addition theorem on probability is used. All the given values are substituted in the addition theorem of probability to find the required probability.

Formula Used: The probability is calculated by,
\[P(E)=\dfrac{n(E)}{n(S)}\]
\[n(E)\] is the number of favorable outcomes and \[n(S)\] is the total number of outcomes.
If there are two events in a sample space, then the addition theorem on probability is given by
\[P(A+B)=P(A)+P(B)-P(A\cap B)\]
In independent events, the occurrence of one event is not affected by the occurrence of another event.
Two events $A$ and \[B\] are said to be independent events if $P(A\cap B)=P(A)\cdot P(B)$ and are said to be mutually exclusive if $P(A\cap B)=\Phi $.

Complete step by step solution: Consider two events $A$ and \[B\].
It is given that,
\[P(A)=0.4\]
\[P(A+B)=0.7\]
\[P(A\cap B)=0.2\]
The addition theorem on probability is given by
\[P(A+B)=P(A)+P(B)-P(A\cap B)\]
Then, by substituting in the formula, we get
\[\begin{align}
  & P(A+B)=P(A)+P(B)-P(A\cap B) \\
 & \text{ 0}\text{.7}=0.4+P(B)-0.2 \\
 & \text{ }\Rightarrow P(B)=0.7-0.2=0.5 \\
\end{align}\]

Option ‘C’ is correct

Note: In this question, the addition theorem on probability is applied for finding the required probability. By substituting the appropriate values, the required probability is calculated.