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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Probability

NDA 2024 · Mathematics · Probability

Let A and B be two events such that P(A B) ≥ 0.75 and 0.125 ≤ P(A...

Let A and B be two events such that P(A∪B)≥0.75and0.125≤P(A∩B)≤0.375\mathrm{P}(\mathrm{A} \cup \mathrm{B}) \geq 0.75 and 0.125 \leq \mathrm{P}(\mathrm{A} \cap \mathrm{B}) \leq 0.375. What is the minimum value of P(A)+P(B)\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B}) ?

Options

  1. A0.625
  2. B0.750
  3. C0.825
  4. D0.875

Correct answer

D. 0.875

Step-by-step solution

Given : P(A∪B)≥0.75\quad \mathrm{P}(\mathrm{A} \cup \mathrm{B}) \geq 0.75 and 0.125≤P(A∩B)≤0.375P(A∪B)≥0.750.125 \leq \mathrm{P}(\mathrm{A} \cap \mathrm{B}) \leq 0.375\mathrm{P}(\mathrm{A} \cup \mathrm{B}) \geq 0.75⇒P(A)+P(B)−P(A∩B)≥0.75⇒P(A)+P(B)≥0.75+P(A∩B)⇒P(A)+P(B)≥0.75+0.125⇒P(A)+P(B)≥0.875\begin{aligned} & \Rightarrow \quad \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \geq 0.75 \\ & \Rightarrow \quad \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B}) \geq 0.75+\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \\ & \Rightarrow \quad \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B}) \geq 0.75+0.125 \\ & \Rightarrow \quad \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B}) \geq 0.875\end{aligned}$MinimumvalueofMinimum value of\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})=0.875$.
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