Let A and B be two independent events of a ra | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Since $A$ and $B$ are independent events,we have $P(A \cap B) = P(A) \cdot P(B)$.Given $P(A \cap B) = \frac{1}{6}$,let $P(A) = x$ and $P(B) = y$.

Vedclass · Accepted Answer

$\frac{1}{2}$ or $\frac{1}{3}$

Vedclass · Answer

(C) Since $A$ and $B$ are independent events,we have $P(A \cap B) = P(A) \cdot P(B)$.Given $P(A \cap B) = \frac{1}{6}$,let $P(A) = x$ and $P(B) = y$. Thus,$xy = \frac{1}{6}$.The probability that neither occurs is $P(A' \cap B') = P((A \cup B)') = 1 - P(A \cup B) = \frac{1}{3}$.Therefore,$P(A \cup B) = 1 - \frac{1}{3} = \frac{2}{3}$.Using the formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$,we get $x + y - \frac{1}{6} = \frac{2}{3}$,which implies $x + y = \frac{5}{6}$.Substituting $y = \frac{5}{6} - x$ into $xy = \frac{1}{6}$,we get $x(\frac{5}{6} - x) = \frac{1}{6}$.This simplifies to $6x^2 - 5x + 1 = 0$.Factoring the quadratic equation,we get $(2x - 1)(3x - 1) = 0$.Thus,$x = \frac{1}{2}$ or $x = \frac{1}{3}$.Hence,the probability of occurrence of $A$ is $\frac{1}{2}$ or $\frac{1}{3}$.

Let $A$ and $B$ be two independent events of a random experiment. If the probability that both $A$ and $B$ occur is $\frac{1}{6}$ and the probability that neither of them occur is $\frac{1}{3}$,then the probability of occurrence of $A$ is

Similar Questions

Two dice are thrown. The probability that the sum of numbers appearing is more than $10$ is:

In each of the following experiments,specify the appropriate sample space: $A$ boy has a $1$ rupee coin,a $2$ rupee coin,and a $5$ rupee coin in his pocket. He takes out two coins out of his pocket,one after the other.

When a die is thrown,let event $A$ be getting a number greater than $3$,and let event $B$ be getting a number less than $5$. What is $P(A \cup B)$?

Two dice are rolled simultaneously. The probability that the sum of the two numbers on the dice is a prime number is:

For two events $A$ and $B$,if $P(A \cup B) = \frac{5}{6}$,$P(A) = \frac{1}{6}$,and $P(B) = \frac{2}{3}$,then $A$ and $B$ are:

Vedclass Test Series

Exam Paper Generator

Online Exam Module