Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Probability

NDA 2024 · Mathematics · Probability

A, B , and C are three events such that P(A)=0.6 , P(B)=0.4,...

A,B\mathrm{A}, \mathrm{B}, and C are three events such that P(A)=0.6\mathrm{P}(\mathrm{A})=0.6, P(B)=0.4,P(C)=0.5,P(A∪B)=0.8,P(A∩C)=0.3\mathrm{P}(\mathrm{B})=0.4, \mathrm{P}(\mathrm{C})=0.5, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=0.8, \mathrm{P}(\mathrm{A} \cap \mathrm{C})=0.3 and P(A∩B∩C)=0.2\mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})=0.2 and P(A∪B∪C)≥0.85\mathrm{P}(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C})\geq 0.85. What is the minimum value of P(B∩C)\mathrm{P}(\mathrm{B} \cap \mathrm{C}) ?

Options

  1. A0.1
  2. B0.2
  3. C0.35
  4. D0.45

Correct answer

B. 0.2

Step-by-step solution

P(A)=0.6,P(B)=0.4,P(C)=0.5P(A∩B)=P(A)+P(B)−P(A∪B)=0.6+0.4−0.8=0.2\begin{aligned} \mathrm{P}(\mathrm{ A})=0.6, \mathrm{P}(\mathrm{ B}) & =0.4, \mathrm{P}(\mathrm{C})=0.5 \\ \mathrm{P}(\mathrm{ A} \cap \mathrm{ B}) & =\mathrm{P}(\mathrm{ A})+\mathrm{P}(\mathrm{ B})-\mathrm{P}(\mathrm{ A} \cup \mathrm{ B}) \\ & =0.6+0.4-0.8=0.2 \end{aligned} Now, P(A∪B∪C)=P(A)+P(B)+P(C)−P(A∪B)−P(A∩C)−P(B∩C)+P(A∩B∩C)=1.5−0.2−0.3−P(B∩C)+0.2\begin{aligned} \mathrm{P}(\mathrm{ A} \cup \mathrm{ B} \cup \mathrm{C})= & \mathrm{P}(\mathrm{ A})+\mathrm{P}(\mathrm{ B})+\mathrm{P}(\mathrm{C}) \\ & -\mathrm{P}(\mathrm{ A} \cup \mathrm{ B})-\mathrm{P}(\mathrm{ A} \cap \mathrm{C}) \\ & -\mathrm{P}(\mathrm{ B} \cap \mathrm{C})+\mathrm{P}(\mathrm{ A} \cap \mathrm{ B} \cap \mathrm{C}) \\ = & 1.5-0.2-0.3-\mathrm{P}(\mathrm{ B} \cap \mathrm{C})+0.2 \end{aligned}⇒P(A∪B∪C)=1.2−P(B∩C)∵0.85≤P(A∪B∪C)≤1⇒0.85≤1.2−P(B∩C)≤1⇒−0.35≤−P(B∩C)≤−0.2⇒0.2≤P(B∩C)≤0.35\begin{array}{rlrl}\Rightarrow & \mathrm{P}(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}) & =1.2-\mathrm{P}(\mathrm{B} \cap \mathrm{C}) \\ \because & 0.85 & \leq \mathrm{P}(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}) \leq 1 \\ \Rightarrow & 0.85 & \leq 1.2-\mathrm{P}(\mathrm{B} \cap \mathrm{C}) \leq 1 \\ \Rightarrow & -0.35 & \leq-\mathrm{P}(\mathrm{B} \cap \mathrm{C}) \leq-0.2 \\ \Rightarrow & 0.2 & \leq \mathrm{P}(\mathrm{B} \cap \mathrm{C}) \leq 0.35\end{array}Minimum value of P\mathrm{P}(B\mathrm{B} \cap C)=0.2\mathrm{C})=0.2.
Practice Probability on Quantrex Academy →

More from Probability

Passage: Four dice are rolled. Question: What is the probability that getting a total of… NDA 2026Three events A , B and C are such that A and B are disjoint, A and C are independent, B… NDA 2026Passage: Let A , B , C and D be mutually exclusive and exhaustive events and… NDA 2026Passage: Four dice are rolled. Question: What is the probability that getting a total of… NDA 2026For a Binomial distribution with mean 6 and standard deviation 2 , what is the value of… NDA 2026Passage: A 4-digit number is selected at random formed by using the digits 0 , 1 , 2 , 3… NDA 2026In an entrance test there are multiple choice questions. There are four options for each… NDA 2026Passage: Let A , B , C and D be mutually exclusive and exhaustive events and… NDA 2026Consider the following statements in respect of the events A , B , C : I. (A B C) ( A B C… NDA 2026Passage: Let A , B , C and D be mutually exclusive and exhaustive events and… NDA 2026The probability that a family owns a laptop is 0.68 ; that it also owns a desktop is 0.56… NDA 2025An urn contains 10 white and 5 red balls. If two balls are drawn at random, then what is… NDA 2025An urn contains 5 white, 6 red and 4 blue balls. Three balls are drawn at random. What is… NDA 2025Under which of the following conditions may binomial distribution be used? I. The number… NDA 2025Three faces of a die are black, two faces are white and one face is red. The die is… NDA 2025A person X speaks the truth 4 out of 5 times and person Y speaks the truth 5 out of 6… NDA 2025 Full Probability list Probability JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs