Question

If events $A$ and \[B\] are two independent, then \[P(A+B)=?\]
A. $P(A)+P(B)-P(A)P(B)$
B. $P(A)-P(B)$
C. $P(A)+P(B)$
D. $P(A)+P(B)+P(A)P(B)$

Answer 1

Hint: In this question, we are to find the condition for the events to be independent. Where two events are said to be independent when the occurrence of one event is not affected by the occurrence of another event. So, using the addition theorem on probability, the required value is calculated.

Formula used: The probability is calculated by,
$P(E)=\dfrac{n(E)}{n(S)}$
Here, the addition theorem on probability is given by
$P(A+B)=P(A)+P(B)-P(A\cap B)$
When two events happen independently, the occurrence of one is not impacted by the occurrence of the other.
For the events $A$ and $B$, $P(A\cap B)=P(A)\cdot P(B)$ if they are independent and $P(A\cap B)=\Phi $ if they are mutually exclusive.

Complete step by step solution: Consider two events $A$ and $B$.
It is given that; they are independent events.
So,
$P(A\cap B)=P(A)\cdot P(B)$
Then, from the addition theorem on probability,
$\begin{align}
  & P(A+B)=P(A)+P(B)-P(A\cap B) \\
 & \text{ =}P(A)+P(B)-P(A)P(B) \\
\end{align}$

Thus, Option (A) is correct.

Note: Here we may go wrong with the value of $P(A\cap B)$. For independent events $P(A\cap B)=P(A)P(B)$. The main formula we use here is the addition theorem on probability. By substituting the appropriate values, the required probability is calculated.