MHT CET · Maths · Probability
Let A and B be two events such that the probability that exactly one of them occurs is \(\frac{2}{5}\) and the probability that A or B occurs is \(\frac{1}{2}\), then the probability of both of them occur together is
- A 0.1
- B 0.2
- C 0.01
- D 0.02
Answer & Solution
Correct Answer
(A) 0.1
Step-by-step Solution
Detailed explanation
Given that,
\(\mathrm{P}\left[\left(\mathrm{~A} \cap \mathrm{~B}^{\prime}\right) \cup\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)\right]=\frac{2}{5}...(i)\)
and \(P(A \cup B)=\frac{1}{2}\)...(ii)
From (i), we get
\(\mathrm{P}(\mathrm{~A})+\mathrm{P}(\mathrm{~B})-2 \mathrm{P}(\mathrm{~A} \cap \mathrm{~B})=\frac{2}{5}...[From(i)]\)
\(\begin{array}{ll}\therefore & P(A \cup B)-P(A \cap B)=\frac{2}{5} \\ \therefore & \frac{1}{2}-P(A \cap B)=\frac{2}{5} \\ \therefore & P(A \cap B)=\frac{1}{2}-\frac{2}{5}=\frac{1}{10}=0.1\end{array}\)
\(\mathrm{P}\left[\left(\mathrm{~A} \cap \mathrm{~B}^{\prime}\right) \cup\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)\right]=\frac{2}{5}...(i)\)
and \(P(A \cup B)=\frac{1}{2}\)...(ii)
From (i), we get
\(\mathrm{P}(\mathrm{~A})+\mathrm{P}(\mathrm{~B})-2 \mathrm{P}(\mathrm{~A} \cap \mathrm{~B})=\frac{2}{5}...[From(i)]\)
\(\begin{array}{ll}\therefore & P(A \cup B)-P(A \cap B)=\frac{2}{5} \\ \therefore & \frac{1}{2}-P(A \cap B)=\frac{2}{5} \\ \therefore & P(A \cap B)=\frac{1}{2}-\frac{2}{5}=\frac{1}{10}=0.1\end{array}\)
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 equation of the plane through the intersection of the planes \(\mathrm{x}+\mathrm{y}+\mathrm{z}=1\) and \(2 \mathrm{x}+3 \mathrm{y}-\mathrm{x}+4=0\) and parallel to \(\mathrm{X}\)-axis isMHT CET 2022 Medium
- A bag contains 5 red balls and 3 green balls. A ball is selected at random and not replaced. A second ball is then selected. The probability of selecting one red ball and one green ball isMHT CET 2022 Medium
- …MHT CET 2019 Medium
- The two vertices of triangle are \((2,-1),(3,2)\) and the third vertex lies on \(x+y=5 .\) The area of the triangle is 4 units, then the third vertex isMHT CET 2012 Easy
- If \(x=\sin \mathrm{t}\) and \(\mathrm{y}=\sin \mathrm{pt}\), then the value of \(\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}+\mathrm{p}^2 \mathrm{y}=\)MHT CET 2025 Medium
- The tangent to the curve \(y=x^3+a x-b\) at the point \((1,-5)\) is perpendicular to the line \(y-x+4=0\), then which one of the following points lies on the curve?MHT CET 2022 Medium
More PYQs from MHT CET
- The shaded area in the figure given below is a solution set of a system of inequations. The minimum value of objective function \(3 x+5 y\), subject to the linear constraints given by this system of inequations is
MHT CET 2023 Easy - Identify the factor from following on which heat of reaction does not depend.MHT CET 2024 Medium
- The value of \(\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}-1=\)MHT CET 2022 Easy
- \(\int \frac{\sin x}{\sqrt{5 \sin ^2 x+6 \cos ^2 x}} \mathrm{~d} x=\)MHT CET 2025 Medium
- Two long straight wires A and B carrying equal current 'I' were kept parallel to each other at distance ' d ' apart. Magnitude of magnetic force experienced by length \(L\) of wire \(A\) is ' \(F\) '. If the distance between the wires is made half and currents are doubled, force \(F_2\) on length \(L\) of wire A will beMHT CET 2025 Medium
- An ideal gas is allowed to expand from \(2 \mathrm{dm}^3\) to \(6 \times 10^{-3} \mathrm{~m}^3\) against a constant external pressure of 1 bar. The work done in \(\mathrm{kJ}\) isMHT CET 2022 Easy