Check whether the following probabilities $P(A)$ and $P(B)$ are consistently defined: $P(A) = 0.5$,$P(B) = 0.7$,$P(A \cap B) = 0.6$.
(N/A) Given: $P(A) = 0.5$,$P(B) = 0.7$,and $P(A \cap B) = 0.6$.
It is a fundamental property of probability that for any two events $A$ and $B$,the intersection of the events must be a subset of each individual event,i.e.,$(A \cap B) \subseteq A$ and $(A \cap B) \subseteq B$.
Consequently,the probability of the intersection must satisfy $P(A \cap B) \leq P(A)$ and $P(A \cap B) \leq P(B)$.
In this case,we observe that $P(A \cap B) = 0.6$ and $P(A) = 0.5$.
Since $0.6 > 0.5$,the condition $P(A \cap B) \leq P(A)$ is violated.
Therefore,the given probabilities $P(A)$ and $P(B)$ are not consistently defined.
It is a fundamental property of probability that for any two events $A$ and $B$,the intersection of the events must be a subset of each individual event,i.e.,$(A \cap B) \subseteq A$ and $(A \cap B) \subseteq B$.
Consequently,the probability of the intersection must satisfy $P(A \cap B) \leq P(A)$ and $P(A \cap B) \leq P(B)$.
In this case,we observe that $P(A \cap B) = 0.6$ and $P(A) = 0.5$.
Since $0.6 > 0.5$,the condition $P(A \cap B) \leq P(A)$ is violated.
Therefore,the given probabilities $P(A)$ and $P(B)$ are not consistently defined.
Explore More
Similar Questions
$A$ coin is tossed and a die is thrown. The probability that the outcome will be a head or a number greater than $4$ or both,is
The probabilities that $A$ and $B$ will die within a year are $p$ and $q$ respectively. Then,the probability that only one of them will be alive at the end of the year is
Medium
View SolutionLet $A$ and $B$ be two events such that $P(\overline{A \cup B}) = \frac{1}{6}$,$P(A \cap B) = \frac{1}{4}$ and $P(\bar{A}) = \frac{1}{4}$,where $\bar{A}$ stands for the complement of the event $A$. Then the events $A$ and $B$ are
MediumAIEEE 2005
View SolutionFrom the word $POSSESSIVE$,a letter is chosen at random. The probability of it being $S$ is
Easy
View SolutionTwo dice are thrown and the sum of the numbers which come up on the dice is noted. Let us consider the following events associated with this experiment:
$A:$ the sum is even.
$B:$ the sum is a multiple of $3$.
$C:$ the sum is less than $4$.
$D:$ the sum is greater than $11$.
Which pairs of these events are mutually exclusive?
$A:$ the sum is even.
$B:$ the sum is a multiple of $3$.
$C:$ the sum is less than $4$.
$D:$ the sum is greater than $11$.
Which pairs of these events are mutually exclusive?
Easy
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo