
If \[4P(A) = 6P(B) = 10P(A \cap B) = 1\] , then find the value of \[P(B\left| {A)} \right.\] .
A. \[\dfrac{2}{5}\]
B. \[\dfrac{3}{5}\]
C. \[\dfrac{7}{{10}}\]
D. \[\dfrac{{19}}{{60}}\]
Answer
164.7k+ views
Hint: First obtain the values of $P(A),P(B),P(A \cap B)$. Then use the formula of
$P(B| {A})$ to obtain the required value.
Formula used: \[P(B\left| {A)} \right. = \dfrac{{P(A \cap B)}}{{P(A)}}\]
Complete step by step solution: The given equation is,
\[4P(A) = 6P(B) = 10P(A \cap B) = 1\]
Therefore,
\[P(A) = \dfrac{1}{4},P(B) = \dfrac{1}{6},P(A \cap B) = \dfrac{1}{{10}}\].
Now,
\[P(B\left| {A)} \right. = \dfrac{{P(A \cap B)}}{{P(A)}}\]
Substitute \[P(A) = \dfrac{1}{4}\] and \[P(A \cap B) = \dfrac{1}{{10}}\] in \[P(B\left| {A)} \right. = \dfrac{{P(A \cap B)}}{{P(A)}}\] to obtain the required value.
\[P(B\left| {A)} \right. = \dfrac{{\dfrac{1}{{10}}}}{{\dfrac{1}{4}}}\]
\[ = \dfrac{4}{{10}}\]
\[ = \dfrac{2}{5}\]
So, Option ‘A’ is correct
Additional information: Bayes' theorem is a mathematical formula used in calculating conditional probability.
The probability can be either conditional or marginal or joint.
Conditional probability: In conditional probability, the probability of an event is dependent on the probability of another event.
At least two dependent events need to apply conditional probability.
Marginal probability: The probability occurring of an event is known as the marginal probability.
Joint probability: Joint is the probability such that two events occur together at the same time.
Note: Students are used to with the formula \[P(A\left| {B)} \right. = \dfrac{{P(A \cap B)}}{{P(B)}}\], so sometimes they divide \[P(A \cap B)\] by \[P(B)\] to obtain the answer but here the question is to find \[P(B\left| {A)} \right.\]. Hence we have to divide \[P(A \cap B)\] by \[P(A)\] to get the correct answer.
$P(B| {A})$ to obtain the required value.
Formula used: \[P(B\left| {A)} \right. = \dfrac{{P(A \cap B)}}{{P(A)}}\]
Complete step by step solution: The given equation is,
\[4P(A) = 6P(B) = 10P(A \cap B) = 1\]
Therefore,
\[P(A) = \dfrac{1}{4},P(B) = \dfrac{1}{6},P(A \cap B) = \dfrac{1}{{10}}\].
Now,
\[P(B\left| {A)} \right. = \dfrac{{P(A \cap B)}}{{P(A)}}\]
Substitute \[P(A) = \dfrac{1}{4}\] and \[P(A \cap B) = \dfrac{1}{{10}}\] in \[P(B\left| {A)} \right. = \dfrac{{P(A \cap B)}}{{P(A)}}\] to obtain the required value.
\[P(B\left| {A)} \right. = \dfrac{{\dfrac{1}{{10}}}}{{\dfrac{1}{4}}}\]
\[ = \dfrac{4}{{10}}\]
\[ = \dfrac{2}{5}\]
So, Option ‘A’ is correct
Additional information: Bayes' theorem is a mathematical formula used in calculating conditional probability.
The probability can be either conditional or marginal or joint.
Conditional probability: In conditional probability, the probability of an event is dependent on the probability of another event.
At least two dependent events need to apply conditional probability.
Marginal probability: The probability occurring of an event is known as the marginal probability.
Joint probability: Joint is the probability such that two events occur together at the same time.
Note: Students are used to with the formula \[P(A\left| {B)} \right. = \dfrac{{P(A \cap B)}}{{P(B)}}\], so sometimes they divide \[P(A \cap B)\] by \[P(B)\] to obtain the answer but here the question is to find \[P(B\left| {A)} \right.\]. Hence we have to divide \[P(A \cap B)\] by \[P(A)\] to get the correct answer.
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

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

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

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets
