KCET · Maths · Probability
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?
- A Statement (I) is true and statement (II) is false
- B Statement (I) is false and statement (II) is true
- C Both the statements are true
- D Both the statements are false
Answer & Solution
Correct Answer
(C) Both the statements are true
Step-by-step Solution
Detailed explanation
\(E\) and \(F\) are two independent events, then \(E^{\prime}\) and \(F^{\prime}\) are also independent. Statement \(I\) true
\(\begin{aligned}
& \mathrm{A} \cap \mathrm{~B}=\phi \Rightarrow \mathrm{P}(\mathrm{~A} \cap \mathrm{~B})=0 \ldots(1) \\
& \mathrm{P}(\mathrm{~A}) \neq 0 \ldots(2) \\
& \mathrm{P}(\mathrm{~B}) \neq 0 \ldots(3)
\end{aligned}\)
From eq(1), (2) and (3)
\(\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \neq \mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B}) \quad\) Statement II is true
\(\begin{aligned}
& \mathrm{A} \cap \mathrm{~B}=\phi \Rightarrow \mathrm{P}(\mathrm{~A} \cap \mathrm{~B})=0 \ldots(1) \\
& \mathrm{P}(\mathrm{~A}) \neq 0 \ldots(2) \\
& \mathrm{P}(\mathrm{~B}) \neq 0 \ldots(3)
\end{aligned}\)
From eq(1), (2) and (3)
\(\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \neq \mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B}) \quad\) Statement II is true
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The set \(\{-1,0,1\}\) is not a multiplicative group because of the failure ofKCET 2008 Easy
- If the area between \(y=m x^{2}\) and \(x=m y^{2}(m>0)\) is \(1 / 4 \mathrm{sq}\) units, then the value of \(m\) isKCET 2011 Hard
- If \(f(x)=a x+b\), where \(a\) and \(b\) are integers, \(f(-1)=-5\) and \(f(\beta)=3\), then \(a\) and \(b\) are respectivelyKCET 2023 Easy
- A stone is thrown vertically upwards from the top of a tower \(64 \mathrm{~m}\) high according to the law \(s=48 t-16 t^{2}\). The greatest height attained by the stone above ground isKCET 2009 Easy
- The value of \( C \) in Mean value theorem for the function \( f(x)=x^{2} \) in \( [2,4] \) isKCET 2017 Easy
- The maximum area of a rectangle that can be inscribed in a circle of radius 2 unit is (in square unit)KCET 2008 Medium
More PYQs from KCET
- The variation of terminal potential difference \( (V) \) with current flowing through as cell is as
shown
The emf and internal resistance of the cell areKCET 2019 Easy - A parallel plate capacitor is charged and then isolated. The effect of increasing the plate separation on charge, potential and capacitance respectively areKCET 2015 Easy
- The enzymes which are absolutely necessary for recombinant DNA technology areKCET 2012 Medium
- \( \int \frac{1}{\sqrt{3-6 x-9 x^{2}}} d x \) is equal toKCET 2018 Easy
- In a series L.C.R circuit, the potential drop across \( L, C \) and \( R \) respectively are \( 40 \mathrm{~V}, 120 \mathrm{~V} \) and \( 60 \mathrm{~V} \). Then the source voltage isKCET 2016 Easy
- During a \(\beta^{-}\)-decayKCET 2020 Medium