Let A and B be two independent events. The pr | vedclass

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

Question

(B) Given $P(A \cap B) = \frac{1}{6}$ and $P(A^c \cap B^c) = \frac{1}{3}$.Since $P(A^c \cap B^c) = P((A \cup B)^c) = 1 - P(A \cup B) = \frac{1}{3}$,we

Vedclass · Accepted Answer

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

Vedclass · Answer

(B) Given $P(A \cap B) = \frac{1}{6}$ and $P(A^c \cap B^c) = \frac{1}{3}$.Since $P(A^c \cap B^c) = P((A \cup B)^c) = 1 - P(A \cup B) = \frac{1}{3}$,we have $P(A \cup B) = \frac{2}{3}$.Using the addition theorem,$P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{2}{3}$.Thus,$P(A) + P(B) = \frac{2}{3} + \frac{1}{6} = \frac{5}{6}$.Since $A$ and $B$ are independent,$P(A \cap B) = P(A)P(B) = \frac{1}{6}$.Let $x = P(A)$ and $y = P(B)$. Then $x + y = \frac{5}{6}$ and $xy = \frac{1}{6}$.These are roots of the quadratic equation $t^2 - (x+y)t + xy = 0$,which is $t^2 - \frac{5}{6}t + \frac{1}{6} = 0$.Multiplying by $6$,we get $6t^2 - 5t + 1 = 0$.$(2t - 1)(3t - 1) = 0$,so $t = \frac{1}{2}$ or $t = \frac{1}{3}$.Therefore,$P(A)$ is $\frac{1}{2}$ or $\frac{1}{3}$.

Let $A$ and $B$ be two independent events. The probability that both $A$ and $B$ occur together is $1/6$ and the probability that neither of them occurs is $1/3$. The probability of occurrence of $A$ is

Similar Questions

If $A$ and $B$ are two events such that $P(A) = 0.4$,$P(A \cup B) = 0.7$ and $P(A \cap B) = 0.2$,then $P(B) = $

If $A$ and $B$ are independent events such that odds in favour of $A$ is $2:3$ and odds against $B$ is $4:5$,then $P(A \cap B)=$

If the probabilities of a student passing in the first,second,or third grade are $1/10$,$3/5$,and $1/4$ respectively,then the probability of the student failing (getting a fourth grade) is ........

Let $A$ and $B$ be two events such that $P(A) = 0.3$ and $P(A \cup B) = 0.8$. If $A$ and $B$ are independent events,then $P(B) = $

If the odds against an event be $2 : 3$,then the probability of its occurrence is

Vedclass Test Series

Exam Paper Generator

Online Exam Module