Consider the following statements.Statement ( | vedclass

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

Question

(C) Statement $(I)$: If $E$ and $F$ are independent,then $P(E \cap F) = P(E)P(F)$. We know that $P(E^{\prime} \cap F^{\prime}) = P((E \cup F)^{\prime}

Vedclass · Accepted Answer

Both the statements are true

Vedclass · Answer

(C) Statement $(I)$: If $E$ and $F$ are independent,then $P(E \cap F) = P(E)P(F)$. We know that $P(E^{\prime} \cap F^{\prime}) = P((E \cup F)^{\prime}) = 1 - P(E \cup F) = 1 - [P(E) + P(F) - P(E \cap F)] = 1 - P(E) - P(F) + P(E)P(F) = (1 - P(E))(1 - P(F)) = P(E^{\prime})P(F^{\prime})$. Thus,$E^{\prime}$ and $F^{\prime}$ are independent. Statement $(I)$ is true.Statement $(II)$: If $E$ and $F$ are mutually exclusive,then $P(E \cap F) = 0$. For them to be independent,we require $P(E \cap F) = P(E)P(F)$. Since $P(E) > 0$ and $P(F) > 0$,$P(E)P(F) > 0$. Thus,$P(E \cap F) 
eq P(E)P(F)$,meaning they cannot be independent. Statement $(II)$ is true.

List-$I$	List-$II$
$A$. $P(\frac{A}{B})$	$(i)$. $\frac{2}{15}$
$B$. $P(\bar{B})$	$(ii)$. $\frac{4}{15}$
$C$. $P(A \cap \bar{B})$	$(iii)$. $\frac{8}{15}$
$D$. $P(B \cap \bar{A})$	$(iv)$. $\frac{2}{3}$
	$(v)$. $\frac{3}{7}$

Consider the following statements.
Statement $(I)$: If $E$ and $F$ are two independent events,then $E^{\prime}$ and $F^{\prime}$ are also independent.
Statement $(II)$: Two mutually exclusive events with non-zero probabilities of occurrence cannot be independent.
Which of the following is correct?

Similar Questions

Let $A, B, C$ be pairwise independent events such that $P(C) > 0$ and $P(A \cap B \cap C) = 0$. Then $P(A' \cap B'|C)$ is equal to:

Prove that if $E$ and $F$ are independent events,then so are the events $E$ and $F^{\prime}$.

If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$,and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$,then the value of $\frac{P(\overline{A} \cup \overline{B})}{P(\overline{A} \cap \overline{B})}$ is:

$A$ couple has two children. Find the probability that both children are males,if it is known that at least one of the children is male.

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Consider the following statements.Statement $(I)$: If $E$ and $F$ are two independent events,then $E^{\prime}$ and $F^{\prime}$ are also independent.Statement $(II)$: Two mutually exclusive events with non-zero probabilities of occurrence cannot be independent.Which of the following is correct?