For two events A and B,if P(A) = P(A|B) = fra | vedclass

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

Question

(D) Given: $P(A) = \frac{1}{4}$,$P(A|B) = \frac{1}{4}$,and $P(B|A) = \frac{1}{2}$.Step $1$: Check for independence.$P(A|B) = \frac{P(A \cap B)}{P(B)}

Vedclass · Accepted Answer

All of the above

Vedclass · Answer

(D) Given: $P(A) = \frac{1}{4}$,$P(A|B) = \frac{1}{4}$,and $P(B|A) = \frac{1}{2}$.Step $1$: Check for independence.$P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{1}{4}$.Also,$P(A|B) = P(A)$ implies that $A$ and $B$ are independent events.Thus,option $A$ is true.Step $2$: Check $P(A'|B)$.Since $A$ and $B$ are independent,$A'$ and $B$ are also independent.Therefore,$P(A'|B) = P(A') = 1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4}$.Thus,option $B$ is true.Step $3$: Check $P(B'|A')$.Since $A$ and $B$ are independent,$A'$ and $B'$ are also independent.Therefore,$P(B'|A') = P(B') = 1 - P(B)$.To find $P(B)$,use $P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{1}{2}$.Since $A, B$ are independent,$P(A \cap B) = P(A)P(B)$.So,$\frac{P(A)P(B)}{P(A)} = P(B) = \frac{1}{2}$.Then $P(B') = 1 - \frac{1}{2} = \frac{1}{2}$.Thus,$P(B'|A') = \frac{1}{2}$,so option $C$ is true.Since $A, B,$ and $C$ are true,the correct answer is $D$.

For two events $A$ and $B$,if $P(A) = P(A|B) = \frac{1}{4}$ and $P(B|A) = \frac{1}{2}$,then which of the following is true?

Similar Questions

If $l, m$ represent any two elements (identical or different) of the set $\{1, 2, 3, 4, 5, 6, 7\}$,then the probability that $lx^2 + mx + 1 > 0$ for all $x \in R$ is

If $A$ and $B$ are two events such that $P(A) = \frac{1}{2}$ and $P(B) = \frac{2}{3},$ then

If $10$ different balls are to be placed in $4$ distinct boxes at random,then the probability that two of these boxes contain exactly $2$ and $3$ balls is

Two dice are thrown and two coins are tossed simultaneously. The probability of getting prime numbers on both the dice along with a head and a tail on the two coins is

$A$ and $B$ are two independent events such that $P(A \cup B) = 0.8$ and $P(A) = 0.3$. The value of $P(B)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module