
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
164.1k+ views
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.
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.
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Main 2025: Conversion of Galvanometer Into Ammeter And Voltmeter in Physics

JEE Advanced 2025 Notes
