
The probabilities of occurrence of two events are respectively $0.21$ and $0.49$. The probability that both occur simultaneously is $0.16$. The probability that neither of the two events occurs is then
A. $0.30$
B. $0.46$
C. $0.14$
D. None of these
Answer
164.7k+ views
Hint: In this question, we are to find the probability of the complement of the union of two events. By using set theory (complement of sets) and addition theorem on probability, the required probability is calculated.
Formula used: According to set theory,
$\begin{align}
& P(\overline{A\cup B})=1-P(A\cup B) \\
& P(\overline{A\cap B})=1-P(A\cap B) \\
\end{align}$
A probability is the ratio of favorable outcomes of an event to the total number of outcomes. So, the probability lies between $0$ and $1$.
The probability is calculated by,
$P(E)=\dfrac{n(E)}{n(S)}$
$n(E)$ - favourable outcomes and $n(S)$ - sample.
If there are two events in a sample space, then the addition theorem on probability is given by
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
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: Given that there are two events.
Consider the two events as $A$ and $B$.
It is given that,
The probability of occurrence the event $A$ is $P\left( A \right)=0.21$
The probability of occurrence the event $B$ is $P\left( B \right)=0.49$
And the probability that both occurs simultaneously is $P\left( A\cap B \right)=0.16$
Then, the probability that none of the two occurs is $P\left( \overline{A\cup B} \right)$
We have
$P(\overline{A\cup B})=1-P(A\cup B)$
By the addition theorem on probability,
$P(A\cup B)=\left[ P(A)+P(B)-P(A\cap B) \right]$
On substituting,
$\begin{align}
& P(\overline{A\cup B})=1-P(A\cup B) \\
& \text{ }=1-\left[ P(A)+P(B)-P(A\cap B) \right] \\
& \text{ }=1-\left[ 0.21+0.49-0.16 \right] \\
& \text{ }=1-0.54=0.46 \\
\end{align}$
Thus, Option (B) 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. Here we may go wrong with the complimented probability. The complement of the events denotes the “odds against” the random experiment.
Formula used: According to set theory,
$\begin{align}
& P(\overline{A\cup B})=1-P(A\cup B) \\
& P(\overline{A\cap B})=1-P(A\cap B) \\
\end{align}$
A probability is the ratio of favorable outcomes of an event to the total number of outcomes. So, the probability lies between $0$ and $1$.
The probability is calculated by,
$P(E)=\dfrac{n(E)}{n(S)}$
$n(E)$ - favourable outcomes and $n(S)$ - sample.
If there are two events in a sample space, then the addition theorem on probability is given by
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
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: Given that there are two events.
Consider the two events as $A$ and $B$.
It is given that,
The probability of occurrence the event $A$ is $P\left( A \right)=0.21$
The probability of occurrence the event $B$ is $P\left( B \right)=0.49$
And the probability that both occurs simultaneously is $P\left( A\cap B \right)=0.16$
Then, the probability that none of the two occurs is $P\left( \overline{A\cup B} \right)$
We have
$P(\overline{A\cup B})=1-P(A\cup B)$
By the addition theorem on probability,
$P(A\cup B)=\left[ P(A)+P(B)-P(A\cap B) \right]$
On substituting,
$\begin{align}
& P(\overline{A\cup B})=1-P(A\cup B) \\
& \text{ }=1-\left[ P(A)+P(B)-P(A\cap B) \right] \\
& \text{ }=1-\left[ 0.21+0.49-0.16 \right] \\
& \text{ }=1-0.54=0.46 \\
\end{align}$
Thus, Option (B) 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. Here we may go wrong with the complimented probability. The complement of the events denotes the “odds against” the random experiment.
Recently Updated Pages
Environmental Chemistry Chapter for JEE Main Chemistry

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

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
