JEE Mains · Maths · STD 12 - 13. probability
Let \(A, B\) and \(C\) be three events, which are pair-wise independence and \(\bar E\) denotes the complement of an event \(E\) . If \(P(A \cap B \cap C) = 0\) and \(P(C) > 0,\) then \(P[(\bar A \cap \bar B)|\,C]\) is equal to
- A \(P(A)\, + \,P(\bar B)\)
- B \(P(\bar A)\, - P(\bar B)\)
- C \(P(\bar A)\, - P(B)\)
- D \(P(\bar A)\, + P(\bar B)\)
Answer & Solution
Correct Answer
(C) \(P(\bar A)\, - P(B)\)
Step-by-step Solution
Detailed explanation
\({\rm{ Here, P}}(\bar A \cap \bar B|{\rm{C}}) = \frac{{P(\bar A \cap \bar B \cap C)}}{{P\left( C \right)}}.\) \( = \frac{{P(C) - P(A \cap C - P(B \cap C) + P(A \cap B \cap C))}}{{P(C)}}\) \(=1-\frac{P(A) \cdot P(C)+P(B) \cdot P(C)}{P(C)} \) \((\because P(A \cap B \cap C)=0)\)…
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
- If \(f(x)=\left|\begin{array}{ccc}2 \cos ^4 x & 2 \sin ^4 x & 3+\sin ^2 2 x \\ 3+2 \cos ^4 x & 2 \sin ^4 x & \sin ^2 2 x \\ 2 \cos ^4 x & 3+2 \sin ^4 x & \sin ^2 2 x\end{array}\right|\) then \(\frac{1}{5} f^{\prime}(0)\) is equal to ...........JEE Mains 2024 Medium
- Let \(A=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]\) and \(P=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right], \theta\gt0\). If \(\mathrm{B}=\mathrm{PAP}^{\mathrm{T}}, \mathrm{C}=\mathrm{P}^{\mathrm{T}} \mathrm{B}^{10} \mathrm{P}\) and the sum of the diagonal elements of \(C\) is \(\frac{\mathrm{m}}{\mathrm{n}}\), where \(\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1\), then \(\mathrm{m}+\mathrm{n}\) is :JEE Mains 2025 Hard
- Let \(O\) be the origin. Let \(\overline{ OP }= x \hat{ i }+ y \hat{ j }-\hat{ k }\) and \(\overline{ OQ }=-\hat{ i }+2 \hat{ j }+3 x \hat{ k }, x , y \in R , x >0,\) be such that \(|\overline{ PQ }|=\sqrt{20}\) and the vector \(\overline{ OP }\) is perpendicular to \(\overline{ OQ }\). If \(\overline{ OR }=3 \hat{ i }+ z \hat{ j }-7 \hat{ k }, z \in R ,\) is coplanar with \(\overline{ OP }\) and \(\overline{ OQ },\) then the value of \(x ^{2}+ y ^{2}+ z ^{2}\) is equal to ...... .JEE Mains 2021 Hard
- The total number of positive integral solutions \(( x , y , z )\) such that \(xyz =24\) isJEE Mains 2021 Hard
- \(2 \sin \left(\frac{\pi}{22}\right) \sin \left(\frac{3 \pi}{22}\right) \sin \left(\frac{5 \pi}{22}\right) \sin \left(\frac{7 \pi}{22}\right) \sin \left(\frac{9 \pi}{22}\right)\) isJEE Mains 2022 Medium
- The function/ defined by \(f(x)\, = x^3 - 3x^2 + 5x + 7\), isJEE Mains 2017 Hard
More PYQs from JEE Mains
- The value of \(\lim _{x \rightarrow 0}\left(\frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}}\right)\) is equal to:JEE Mains 2021 Hard
- Let \(f(x)=\int_{0}^{x} e^{t} f(t) d t+e^{x}\) be a differentiable function for all \(x \in R\). Then \(f(x)\) equals ..... .JEE Mains 2021 Hard
- Let \(p_n\) denote the total number of triangles formed by joining the vertices of an \(n\)-side regular polygon. If \(p_{n+1} - p_n = 66\), then the sum of all distinct prime divisors of \(n\) is:JEE Mains 2026 Medium
- The inverse function of \(f(\mathrm{x})=\frac{8^{2 \mathrm{x}}-8^{-2 \mathrm{x}}}{8^{2 \mathrm{x}}+8^{-2 \mathrm{x}}}, \mathrm{x} \in(-1,1),\) isJEE Mains 2020 Hard
- Let \(A\) be a \(2 \times 2\) matrix with \(\operatorname{det}(A)=-1\) and det \((( A + I )(\operatorname{Adj}( A )+ I ))=4\). Then the sum of the diagonal elements of \(A\) can be.JEE Mains 2022 Hard
- Let the function \(f(x)=\left(x^2+1\right)\left|x^2-a x+2\right|+\cos |x|\) be not differentiable at the two points \(x=\alpha=2\) and \(x=\beta\). Then the distance of the point \((\alpha, \beta)\) from the line \(12 x+5 y+10=0\) is equal to :JEE Mains 2025 Hard