Let A, B, C be three pairwise independent eve | vedclass

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

Question

(C) Given that $A, B, C$ are pairwise independent events, we have $P(A \cap B) = P(A)P(B)$, $P(B \cap C) = P(B)P(C)$, and $P(C \cap A) = P(C)P(A)$.We

Vedclass · Accepted Answer

$\frac{3}{2}-b-c$

Vedclass · Answer

(C) Given that $A, B, C$ are pairwise independent events, we have $P(A \cap B) = P(A)P(B)$, $P(B \cap C) = P(B)P(C)$, and $P(C \cap A) = P(C)P(A)$.We are given $P(\bar{B} \cup \bar{C}) = \frac{1}{2}$.By De Morgan's Law, $P(\bar{B} \cup \bar{C}) = 1 - P(B \cap C) = \frac{1}{2}$, which implies $P(B \cap C) = \frac{1}{2}$.Since $B$ and $C$ are independent, $P(B)P(C) = bc = \frac{1}{2}$.We need to find $P((\bar{B} \cap \bar{C}) \mid A) = \frac{P((\bar{B} \cap \bar{C}) \cap A)}{P(A)}$.Using the property of sets, $(\bar{B} \cap \bar{C}) \cap A = A \setminus (A \cap (B \cup C)) = A \setminus ((A \cap B) \cup (A \cap C))$.Thus, $P((\bar{B} \cap \bar{C}) \cap A) = P(A) - P((A \cap B) \cup (A \cap C)) = P(A) - [P(A \cap B) + P(A \cap C) - P(A \cap B \cap C)]$.Since the events are pairwise independent, $P(A \cap B \cap C)$ is not necessarily $P(A)P(B)P(C)$ unless they are mutually independent. However, the question implies pairwise independence. Assuming mutual independence for the intersection term:$P((\bar{B} \cap \bar{C}) \cap A) = P(A) - P(A)P(B) - P(A)P(C) + P(A)P(B)P(C) = P(A)(1 - b - c + bc)$.Substituting $bc = \frac{1}{2}$, we get $P((\bar{B} \cap \bar{C}) \cap A) = P(A)(1 - b - c + \frac{1}{2}) = P(A)(\frac{3}{2} - b - c)$.Therefore, $P((\bar{B} \cap \bar{C}) \mid A) = \frac{3}{2} - b - c$.

Let $A, B, C$ be three pairwise independent events of a random experiment. If $P(\bar{B} \cup \bar{C}) = \frac{1}{2}$, $P(A) > 0$, $P(B) = b$, and $P(C) = c$, then $P((\bar{B} \cap \bar{C}) \mid A) = $

Similar Questions

$A$ and $B$ throw a die alternatively till one of them gets a '$6$' and wins the game. Find their respective probabilities of winning,if $A$ starts first.

Two balls are drawn from an urn containing $7$ white,$6$ red,and $8$ black balls one after the other without replacement. The probability that at least one of them is white is:

The probability that a mechanic makes an error while using a machine on the $n$th day is given by $P(E_n) = \frac{1}{2^n}$. If he has operated the machine for $4$ days, the probability that he has not made a mistake on $3$ of the $4$ days is:

$A$ fair die is rolled indefinitely. Player $A$ wins if two consecutive rolls show $3$ or $5$, and player $B$ wins if two consecutive rolls show $1$ or $2$ or $4$ or $6$. The probability that player $A$ wins is:

If $n$ numbers are chosen at random from the set $\{1, 2, 3, \dots, 1000\}$,what is the probability that $\frac{\sum_{i=1}^n i^2}{\sum_{i=1}^n i}$ is an integer?

Vedclass Test Series

Exam Paper Generator

Online Exam Module